/* ident04.gp -- Special Algebraic Identities */
/* 31 Jan 2006 Michael Somos */
/* 21 Jun 2024 latest version */
/* This code is for the GPL computer algebra system PARI-GP (version 2). */
/* Please see "http://pari.math.u-bordeaux.fr/" for more information */
/* The minimum runtime stack required. You may want more */
NN = 4*10^6; /* need more memory for id7_7_1_15 */
/* This is to avoid stack overflow while computing */
if(default(parisize) a+b, b -> a-b} in id2_3_1_2a.
\\ Limiting case of identity: hav(a+b) - hav(a-b) = sin(a)*sin(b).
\\ {} 2 vars with 3 terms highest degree 1 of total 2 [TS]
{ id2_3_1_2b = +(a-b)*(a-b) -(a+b)*(a+b) +(a+a)*(b+b) ; }
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\\ This is a rare identity notable for having two tags.
\\ {} 2 vars with 3 terms highest degree 1 of total 2 [TS,NC]
{ id2_3_1_2c = +a*(a+b) -a*(a-b) -(a+a)*b ; }
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\\ {} 2 vars with 3 terms highest degree 1 of total 2 [TS]
{ id2_3_1_2d = +a*(a+a) +(a-b)*b -(a+b)*(a+a-b) ; }
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\\ {} 2 vars with 3 terms highest degree 1 of total 2 [TS]
{ id2_3_1_2e = +a*(b+b-a) +(a-b)*(a+b) -b*(a+a-b) ; }
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\\ {} 2 vars with 3 terms highest degree 1 of total 3 [TS,CT]
{ id2_3_1_3a = +2*a*b*(a+b) -(a+a)*b*b -a*a*(b+b) ; }
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\\ {} 2 vars with 3 terms highest degree 1 of total 3
{ id2_3_1_3b = +a*a*(a+a) -(a+b)*(a-b-b)*(a-b-b) -(a-b)*(a+b+b)*(a+b+b) ; }
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\\ A two variable functional equation for the Weierstrass sigma function.
\\ {} 2 vars with 3 terms highest degree 1 of total 4 [WS]
{ id2_3_1_4a = +a*a*a*(b+b-a) -(a+a-b)*b*b*b +(a-b)*(a-b)*(a-b)*(a+b) ; }
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\\ A two variable functional equation for the Weierstrass sigma function.
\\ {} 2 vars with 3 terms highest degree 1 of total 4 [WS]
{ id2_3_1_4b = +a*(a+b)*(a+b)*(a-b-b) -a*(a-b)*(a-b)*(a+b+b)
+b*b*(a+a)*(b+b) ; }
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\\ A two variable functional equation for the Weierstrass sigma function.
\\ {} 2 vars with 3 terms highest degree 1 of total 4 [WS]
{ id2_3_1_4c = +a*a*(a+b)*(a-b-b-b) -(a-b)*(a-b)*(a+b+b)*(a-b-b)
+(a+a-b)*b*(b+b)*(b+b) ; }
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\\ Suppose that A^3 + B^3 = k*C^3. Then (A,B,C) is a point on a homogeneous
\\ elliptic curve. Now let a = A^3, b = B^3, X = A*(a+b+b), Y = -B*(a+a+b),
\\ Z = C*(a-b). Then X^3 + Y^3 = k*Z^3 is a duplication formula.
\\ Replace {a -> a+2*b, b -> 2*a+b} in id2_3_1_4a.
\\ {} 2 vars with 3 terms highest degree 1 of total 4
{ id2_3_1_4d = +a*(a+b+b)*(a+b+b)*(a+b+b) -b*(a+a+b)*(a+a+b)*(a+a+b)
-(a+b)*(a-b)*(a-b)*(a-b) ; }
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\\ A two variable functional equation for the Weierstrass sigma function.
\\ {} 2 vars with 3 terms highest degree 1 of total 4 [WS]
{ id2_3_1_4e = +a*a*b*(b+b+b) +b*b*(a+b+b)*(a-b-b)
-(b+b)*(b+b)*(a+b)*(a-b) ; }
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\\ This is the sum of two squares expressed as difference of two rectangles.
\\ {} 2 vars with 3 terms highest degree 2 of total 2 [NC]
{ id2_3_2_2a = +(a*a+b*b) -(a+b)*a +b*(a-b) ; }
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\\ {} 2 vars with 3 terms highest degree 2 of total 2 [CT]
{ id2_3_2_2b = +(a*a+b*b) +(a+b)*(a+b) -2*(a*a+a*b+b*b) ; }
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\\ case n=3 of 0 = a^n - b^n - (a-b)*(a^(n-1) + ... + b^(n-1)).
\\ {} 2 vars with 3 terms highest degree 2 of total 3
{ id2_3_2_3a = +a*a*a -b*b*b -(a-b)*(a*a+a*b+b*b) ; }
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\\ {} 2 vars with 3 terms highest degree 2 of total 3
{ id2_3_2_3b = +a*a*a -b*(a*a+a*b-b*b) -(a-b)*(a-b)*(a+b) ; }
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\\ {} 2 vars with 3 terms highest degree 2 of total 3
{ id2_3_2_3c = +a*a*a +(a+b)*(a+b)*(a+b) -(a+a+b)*(a*a+a*b+b*b) ; }
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\\ {} 2 vars with 3 terms highest degree 2 of total 4
{ id2_3_2_4a = +a*a*b*b +(a+b)*(a+b)*(a*a+b*b) -(a*a+a*b+b*b)*(a*a+a*b+b*b) ; }
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\\ {} 2 vars with 3 terms highest degree 2 of total 4
{ id2_3_2_4b = +a*a*a*a -b*b*b*b -(a-b)*(a+b)*(a*a+b*b) ; }
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\\ {} 2 vars with 3 terms highest degree 2 of total 4
{ id2_3_2_4c = +a*a*a*a -b*(a-b)*(a+b)*(a+a+b) -(a*a-a*b-b*b)*(a*a-a*b-b*b) ; }
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\\ This comes from a well known Pythagorean triple parametrization.
\\ {} 2 vars with 3 terms highest degree 2 of total 4
{ id2_3_2_4d = +(a-b)*(a-b)*(a+b)*(a+b) +a*(a+a)*b*(b+b) -(a*a+b*b)*(a*a+b*b) ; }
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\\ This is Sophie Germain" s {} 2 vars 3 4 id2_3_2_4e="+a*a*a*a" +4*b*b*b*b -(a*a+b*b+b*b-a*b-a*b)*(a*a+b*b+b*b+a*b+a*b) similar pythagorean triple parametrization. u^2 v^2="2w^2." id2_3_2_4f="+2*(a*a+b*b)*(a*a+b*b)" -(a*a-a*b-a*b-b*b)*(a*a-a*b-a*b-b*b) -(a*a+a*b+a*b-b*b)*(a*a+a*b+a*b-b*b) 6 id2_3_2_6="+a*a*a*a*a*a" -b*b*b*b*b*b -(a-b)*(a+b)*(a*a+a*b+b*b)*(a*a-a*b+b*b) equivalent s4*s2^3 s3^3*s1 s1="sin(x)," s2="sin(2*x)," ..., s5="sin(5*x)" b="exp(-i*x)." id2_3_4_6="+a*b*(a*a*a*a+a*a*a*b+a*a*b*b+a*b*b*b+b*b*b*b)" +(a*a+a*b+b*b)*(a*a+a*b+b*b)*(a*a+a*b+b*b) -(a+b)*(a+b)*(a+b)*(a+b)*(a*a+b*b) s5*s3*s2^2 s4^2*s3*s1 s6="sin(6*x)" 5 8 id2_3_5_8="+a*b*(a+b)*(a*a*a*a*a+a*a*a*a*b+a*a*a*b*b+a*a*b*b*b+a*b*b*b*b+b*b*b*b*b)" +(a*a*a+a*a*b+a*b*b+b*b*b)*(a*a*a+a*a*b+a*b*b+b*b*b)*(a*a+a*b+b*b) -(a+b)*(a+b)*(a*a*a*a+a*a*a*b+a*a*b*b+a*b*b*b+b*b*b*b)*(a*a+a*b+b*b) cyclotomic (1-x)*(1-x^6) x*(1-x)*(1-x^2)*(1-x^3). 7 id2_3_6_7="+a*a*(a*a-b*b)*(a*a*a-b*b*b)" -(a-b)*(a*a*a*a*a*a-b*b*b*b*b*b) -b*(a-b)*(a*a-b*b)*(a*a*a-b*b*b) split first term id2_3_2_2 into variation. step induction f(n+1)f(n-1)="F(n)^2" (-1)^n 1 id2_4_1_2a="+a*a" +b*b -(a+b)*a +b*(a-b) id2_4_1_2b="+a*a" -a*b id2_4_1_2c="+a*(b-a)" +(b-a)*b +a*(a+b) -(a+b)*b [ts,nc] id2_4_1_2d="+a*a" +a*(b+b-a) -(a+a-b)*b id2_4_1_2e="+a*b" +a*(a+a+b) -b*(a+b) -(a+a-b)*(a+b) [ct,nc] id2_4_1_2f="+2*a*a" +2*b*b -(a+b)*(a+b) -(a-b)*(a-b) id2_4_1_2g="+a*a" -b*(a-b) generating pell's $x^2-2y^2="\pm1$" id2_4_1_2h="+a*a" -2*(a+b)*(a+b) +(a+b+b)*(a+b+b) -2*b*b id2_4_1_2i="+a*a" +a*(a+b+b) -(a-b)*(a+b) id2_4_1_3a="+a*a*a" +b*b*b -a*b*(a+b) -(a+b)*(a-b)*(a-b) id2_4_1_3b="+a*b*(a+b)" -a*(a-b)*(a+a-b) +(a+a)*(a+b)*(a-b) -(a+a)*b*(a+a-b) id2_4_1_3c="+a*a*a" -a*b*b -a*(a+b)*(a+b) +(a+a)*(a+b)*b id2_4_1_3d="+a*a*a" +3*a*b*(a+b) -(a+b)*(a+b)*(a+b) id2_4_1_3e="+8*a*a*a" +3*a*a*b +b*(a+a+a+b)*(a+a+a+b) -(a+a+b)*(a+a+b)*(a+a+b) id2_4_1_3f="+a*b*(a+b)" +a*(a+b)*(a+a+b) -a*a*(a+a) -(a+a)*b*(a+a+b) id2_4_1_4="+a*a*(a-b-b)*(a-b-b)" -(a+a-b)*(a+a-b)*b*b +(a-b)*(a-b)*(a+b)*(a+b) -(a+a)*(a-b)*(a+b)*(a-b-b) id2_4_1_6="+a*a*a*a*a*(b+b-a)" -(a+a-b)*b*b*b*b*b +(a-b)*(a-b)*(a-b)*(a-b)*(a-b)*(a+b) -a*b*(a-b)*(a+b)*(a+a-b)*(b+b-a) gabriel berger, "relations between cusp forms congruence noncongruence groups", proc. ams, v. 128 (2000) pp. 2869-2874, (p. 2872, equ. (10)): "[[...]] (\frac{1}{x^3+x} +\frac{1}{x^3-x} -\frac{1}{(x^2+1)(x-1)} -\frac{1}{(x^2+1)(x+1)}). identically zero." id2_4_2_2a="+(a*a+b*b)" -a*(a-b) +(a+b)*(a-b) id2_3_2_2b id2_4_2_2b="+a*a" +(a+b)*(a+b) -2*(a*a+a*b+b*b) id2_4_2_3a="+a*a*a" +a*b*(a+b) -(a+b)*(a*a+b*b) id2_4_2_3b="+a*a*a" +2*a*b*(a+b) -(a+b)*(a*a+a*b+b*b) id2_4_2_4a="+a*a*a*(a-b)" -a*(a-b)*(a*a-a*b+b*b) +(a*a+b*b)*(a*a+b*b) -(a+b)*(a+b)*(a*a-a*b+b*b) leonard e. dickson, history theory numbers, vol. 2, 266) : "u. dainelli derived integration $c="0$" catalan's formula $$ (a^2+ab+b^2)^2="(ab)^2" +{a(a+b)}^2 {b(a+b)}^2. [[...]]". id2_4_2_4b="+a*a*b*b" +(a+b)*(a+b)*b*b +(a+b)*(a+b)*a*a -(a*a+a*b+b*b)*(a*a+a*b+b*b) from wikipedia, "candido's identity, named after italian mathematician giacomo candido, real numbers. ... can interpreted geometrically. states area side lengths x^2+y^2+(x+y)^2 equals twice areas squares x^2, y^2 (x+y)^2." 657) "f. proth recalled whence $ 2(a^2+ab+b^)^2="a^4+b^4+(a+b)^4." $" 659) "a. cunningham miot obtained x^4+y^4+(x+y)^4="2(x^2+xy+y^2)^2." 702) discussed a^4+b^4+(a+b)^4="2(a^2+ab+b^)^2." id2_4_2_4c="+a*a*a*a" +b*b*b*b +(a+b)*(a+b)*(a+b)*(a+b) -2*(a*a+a*b+b*b)*(a*a+a*b+b*b) id2_4_2_4d="+a*a*a*a" +a*a*b*b -(a*a+a*b+b*b)*(a*a-a*b+b*b) id2_4_2_4e="+a*a*a*a" -2*(a*a+a*b+b*b)*(a*a-a*b+b*b) id2_4_2_5a="+a*a*a*(a*a+b*b)" -a*a*a*(a*a+a*b+b*b) +b*(a-b)*(a+b)*(a*a+b*b) +b*b*b*b*b 752) "... right member initial becomes $5xy(x+y)(x^2+xy+y^2)^2$ $n="7$" ..." confusion 7. id2_4_2_5b="+(a+b)*(a+b)*(a+b)*(a+b)*(a+b)" -a*a*a*a*a -b*b*b*b*b -5*a*b*(a+b)*(a*a+a*b+b*b) bruce reznick jeremy rouse, "sum cubes", (4.13): "(x^2+xy-y^2)^3 (x^2-xy-y^2)^3="2x^6" 2y^6" id2_4_2_6="+2*a*a*a*a*a*a" -2*b*b*b*b*b*b -(a*a+a*b-b*b)*(a*a+a*b-b*b)*(a*a+a*b-b*b) -(a*a-a*b-b*b)*(a*a-a*b-b*b)*(a*a-a*b-b*b) 738) setting $a="a^7," r="a+b," c="ab,$" get (a+b)^7-a^7-b^7="7ab(a+b)(a^2+ab+b^2)^2." id2_4_2_7="+(a+b)*(a+b)*(a+b)*(a+b)*(a+b)*(a+b)*(a+b)" -a*a*a*a*a*a*a -b*b*b*b*b*b*b -7*a*b*(a+b)*(a*a+a*b+b*b)*(a*a+a*b+b*b) mathworld, diophantineequation3rdpowers (6) vieta cubes: x^3 y^3="(x(x^3+2y^3)/(x^3-y^3))^3" (y(y^3+2x^3) (y^3-x^3))^3 12 id2_4_3_12="-a*a*a*(a*a*a+b*b*b)*(a*a*a+b*b*b)*(a*a*a+b*b*b)" +a*a*a*(a*a*a-b*b*b-b*b*b)*(a*a*a-b*b*b-b*b*b)*(a*a*a-b*b*b-b*b*b) +b*b*b*(a*a*a+b*b*b)*(a*a*a+b*b*b)*(a*a*a+b*b*b) +b*b*b*(a*a*a+a*a*a-b*b*b)*(a*a*a+a*a*a-b*b*b)*(a*a*a+a*a*a-b*b*b) id2_4_4_6="+(a*a+a*b+b*b)*(a*a*a*a+b*b*b*b)" -(a*a-a*b+b*b)*(a*a*a*a+b*b*b*b) +(a*a+b*b)*(a*a+b*b)*(a*a+a*b+b*b) -(a+b)*(a+b)*(a+b)*(a+b)*(a*a-a*b+b*b) [tt,ct] id2_5_1_2="+a*(a+b)" +a*(a-b) -2*(a+b)*(a-b) [tt,ct,zs] id2_5_1_3a="+a*a*(a+a)" -a*b*(a+a) -a*(a+a)*(a-b) -(a+a)*b*(a-b) +2*a*b*(a-b) id2_5_1_3b="+a*a*(a+a)" -a*a*(a-b) -a*a*(a+b) -(a+a)*(a-b)*(a+b) +2*a*(a-b)*(a+b) id2_5_1_3c="+a*a*a" -a*(a-b)*b -(a-b)*b*b -b*b*b id2_6_1_2="+a*b" -a*(b+b) +(a+b)*b -(a-b)*b +(a-b-b)*(b+b) -(a-b-b)*b [je,zs] id2_6_1_4="+a*a*(a+b)*(a+b+b)" +a*a*(a-b)*(a+b+b) -a*(a-b)*(a+b)*(a+b) +a*b*b*(a-b) -b*b*(a+b)*(a+b+b) -(a-b)*(a+b)*(a+b)*(a+b+b) given seven consecutive [w1, w2, w3, w4, w5, w6 ,w7] general somos-4 sequence, determinant 3x3 rank matrix [w1*w5, w2*w4, w3*w3; w2*w6, w3*w5, w4*w4; w3*w7, w4*w6, w5*w5]. replace {w1> a-b-b-b, w2 -> a-b-b, w3 -> a-b, w4 ->a, w5 -> a+b,
\\ w6 -> a+b+b, w7 -> a+b+b+b}.
\\ {} 2 vars with 6 terms highest degree 1 of total 6 [WS]
{ id2_6_1_6 = +(a-b-b-b)*(a-b)*(a+b)*(a+b)*(a+b)*(a+b)
-(a-b-b-b)*a*a*a*(a+b)*(a+b+b)
-(a-b-b)*(a-b-b)*a*(a+b)*(a+b)*(a+b+b)
+(a-b-b)*(a-b)*(a-b)*a*(a+b+b)*(a+b+b)
+(a-b-b)*(a-b)*a*a*a*(a+b+b+b)
-(a-b)*(a-b)*(a-b)*(a-b)*(a+b)*(a+b+b+b) ; }
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\\ {} 2 vars with 7 terms highest degree 1 of total 3 [TT,CT]
{ id2_7_1_3 = +a*(a+b)*(a+b) -b*(a+b)*(a+b) -a*(a-b)*(a-b) -b*(a-b)*(a-b)
+2*b*(a-b)*(a+b) -2*(a-b)*(a+b)*(a+b) +2*(a-b)*(a-b)*(a+b) ; }
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\\ {} 2 vars with 8 terms highest degree 1 of total 3 [TT,CT,ZS]
{ id2_8_1_3a = +a*a*a +a*a*b -a*a*(a+b) -3*a*a*(a+a+a) +3*a*b*(a+b)
-3*a*b*(a+a+a) +3*a*(a+a+a)*(a+b) -b*(a+a+a)*(a+b) ; }
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\\ {} 2 vars with 8 terms highest degree 1 of total 3 [TT,CT]
{ id2_8_1_3b = +a*a*(a+b) +a*a*(a-b) -3*a*(a-b)*(a+b) -3*a*(a+b)*(a+b+b)
+a*(a+b)*(a+b) +a*(a-b)*(a+b+b) +(a+b)*(a+b)*(a+b+b)
+(a-b)*(a+b)*(a+b+b) ; }
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\\ {} 2 vars with 8 terms highest degree 1 of total 4 [JE]
{ id2_8_1_4 = +(a-b-b)*(a-b)*(a-b)*(a+b) -(a-b-b)*(a-b)*a*a
-(a-b-b)*(a-b)*a*(a+b+b) +(a-b-b)*a*(a+b)*(a+b+b)
+(a-b)*(a-b)*a*(a+b) -(a-b)*a*(a+b)*(a+b)
-(a-b)*(a+b)*(a+b)*(a+b+b) +a*a*(a+b)*(a+b+b) ; }
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\\ {} 2 vars with 12 terms highest degree 1 of total 4 [WZ,CT,ZS]
{ id2_12_1_4 = +a*b*(a-b)*(a+b) +a*b*(a-b)*(a-b-b) -a*b*(a-b)*(a+a-b)
-a*b*(a+b)*(a-b-b) -a*b*(a+b)*(a+a-b) -a*(a-b)*(a-b-b)*(a+a-b)
-b*(a-b)*(a-b-b)*(a+a-b) -3*a*(a+b)*(a-b-b)*(a+a-b)
+3*b*(a+b)*(a-b-b)*(a+a-b) +3*(a-b)*(a+b)*(a-b-b)*(a+a-b)
+a*(a-b)*(a+b)*(a+a-b) -b*(a-b)*(a+b)*(a-b-b) ; }
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\\ {} 2 vars with 12 terms highest degree 1 of total 7 [WZ,CT,ZS]
{ id2_12_1_7 =
+2*a*a*b*b*(a+a)*(b+b)*(a+b)
-a*a*b*b*(a+a)*(b+b)*(a+a+b+b)
+2*a*a*(a+a)*b*(a+b)*(a+b)*(a+a+b+b)
-a*a*(a+a)*(b+b)*(a+b)*(a+b)*(a+a+b+b)
+2*a*b*b*(b+b)*(a+b)*(a+b)*(a+a+b+b)
-(a+a)*b*b*(b+b)*(a+b)*(a+b)*(a+a+b+b)
-a*a*b*b*(a+a)*(a+b)*(a+b)
-a*a*b*b*(b+b)*(a+b)*(a+b)
+a*a*b*b*(a+b)*(a+b)*(a+a+b+b)
-2*a*a*b*(a+a)*(b+b)*(a+b)*(a+a+b+b)
-2*a*b*b*(a+a)*(b+b)*(a+b)*(a+a+b+b)
+2*a*b*(a+a)*(b+b)*(a+b)*(a+b)*(a+a+b+b)
; }
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\\ {} 2 vars with 14 terms highest degree 1 of total 6 [WZ,CT,ZS]
{ id2_14_1_6 =
+a*a*(a+a)*b*(b-a)*(b+a)
-a*a*(a+a)*b*(b-a)*(a+a+a)
-a*a*(a+a)*b*(b-a)*(b-a-a)
+a*a*b*(b-a)*(b-a-a)*(b+a)
+a*(a+a)*b*(b-a)*(b-a-a)*(b+a)
-3*a*(a+a+a)*b*(b-a)*(b-a-a)*(b+a)
+(a+a)*(a+a+a)*b*(b-a)*(b-a-a)*(b+a)
+a*a*(a+a)*(a+a+a)*b*(b+a)
-a*a*(a+a)*(a+a+a)*(b-a-a)*(b+a)
+2*a*(a+a)*(a+a+a)*(b-a-a)*b*(b+a)
-a*(a+a)*(a+a+a)*b*(b-a)*(b+a)
+a*a*(a+a)*(a+a+a)*(b-a)*(b-a-a)
+a*(a+a)*(a+a+a)*b*(b-a)*(b-a-a)
-2*a*(a+a)*(a+a+a)*(b-a)*(b-a-a)*(b+a)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 2 vars with 32 terms highest degree 1 of total 3 [JZ,CT,ZS]
{ id2_32_1_3 =
+b*b*b
-a*b*b
-2*a*b*(a-b)
-b*b*(a-b)
+a*(b+b)*(a-b)
-a*b*(b+b)
+2*b*(b+b)*(a-b)
+b*b*(b+b)
-a*b*(b+b+b)
-(a-b)*b*(b+b+b)
-3*b*b*(b+b+b)
+a*(b+b)*(b+b+b)
+b*(b+b)*(b+b+b)
-a*b*(a+b)
+3*(a-b)*b*(a+b)
+b*b*(a+b)
-a*(b+b)*(a+b)
+b*(b+b)*(a+b)
+a*(b+b+b)*(a+b)
-(a-b)*(b+b+b)*(a+b)
+b*(b+b+b)*(a+b)
-(b+b)*(b+b+b)*(a+b)
+3*a*b*(a+b+b)
-b*(a-b)*(a+b+b)
+b*b*(a+b+b)
-(a-b)*(b+b)*(a+b+b)
-2*b*(b+b)*(a+b+b)
-a*(b+b+b)*(a+b+b)
+(a-b)*(b+b+b)*(a+b+b)
+b*(b+b+b)*(a+b+b)
-2*b*(a+b)*(a+b+b)
+(b+b)*(a+b)*(a+b+b)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 3 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a directed sum of three sides of a triangle.
\\ case n=3 of (x_1-x_n) = (x_1-x_2) + (x_2-x_3) + ... + (x_{n-1}-x_n).
\\ case n=1 of 0 = (x_1-y_1)(x_2-y_2)...(x_n-y_n) +
\\ (y_1-z_1)*(x_2-y_2)...(x_n-y_n) + (x_1-z_1)*(y_2-z_2)*(x_3-y_3)...(x_n-y_n)
\\ + ... + (x_1-z_1)*(x_2-z_2)...(y_n-z_n) - (x_1-z_1)*(x_2-z_2)...(x_n-z_n).
\\ B. C. Berndt, Ramanujan's Notebooks, Part IV, (p. 36., (25.1)) case n=1 :
\\ "$$ \sum_{k=0}^{n-1} \frac{a_0a_1\dots a_k}{(x+a_1)(x+a_2)\dots(x+a_{k+1})}
\\ = \frac{1}{x}-\frac{1}{x(1+x/a_1)(1+x/a_2)\dots(1+x/a_n)}. $$"
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=3, r=0 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 3 vars with 3 terms highest degree 1 of total 1 [NC]
{ id3_3_1_1a = +(a-b) +(b-c) -(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 1 of total 1 [NC]
{ id3_3_1_1b = +(a+b-c-c) +(a-b-b+c) -(a+a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ T. Amdeberhan, O. Espinosa, V. Moll, A. Straub, "Wallis-Ramanujan-Schur-
\\ Feynman", (p.5, equ. (4.3)) case n=2 : "\prod_{k=1}^n \frac{1}{y+b_k} =
\\ \sum_{k=1}^n \frac{1}{y+b_k}\prod_{j=1,j\ne k}^n\frac{1}{b_j-b_k}. "
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=3, r=1 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 3 vars with 3 terms highest degree 1 of total 2 [TS]
{ id3_3_1_2a = +a*(b-c) -b*(a-c) +c*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {a -> (-a+b+c)/2, b -> (a-b+c)/2, c -> (a+b-c)/2} to get id3_3_1_2a.
\\ {} 3 vars with 3 terms highest degree 1 of total 2 [TS]
{ id3_3_1_2b = +(a-b)*(a+b) -(a-c)*(a+c) +(b-c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {c -> a-c} to get id3_3_1_2a.
\\ Pfaff's 1797 Hypergeometric _3F_2 sum case n=1 :
\\ Vajda's identity is $ F_{n+i}F_{n+j} - F_nF_{n+i+j} = (-1)^nF_iF_j $ for
\\ Fibonacci numbers. Replace {n -> a, j -> -b, i -> -c}.
\\ Arthur Cayley, A Trigonometrical Identity, Messenger of Math., 7 (1878),
\\ p. 124. "cos(b-c) cos(b+c+d) + cos a cos(a+d) = cos(c-a) cos(c+a+d) +
\\ cos b cos(b+d) = cos(a-b) cos(a+b+d) + cos c cos(c+d) = cos a cos(a+d) +
\\ cos b cos(b+d) + cos c cos(c+d) - cos d." To get identity id3_3_1_2c first
\\ replace {a -> a-pi/2, d -> -b-c} in "cos(a-b) cos(a+b+d) + cos c cos(c+d)
\\ = cos a cos(a+d) + cos b cos(b+d) + cos c cos(c_d) - cos d", then use the
\\ identity "cos(b+c) = cos(b) cos(c) - sin(b) sin(c)", and then take limits.
\\ {} 3 vars with 3 terms highest degree 1 of total 2 [TS]
{ id3_3_1_2c = +(a-b)*(a-c) -b*c -a*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 1 of total 2 [TS]
{ id3_3_1_2d = +(a-b)*(a+b-c) -a*(a-c) +b*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 1 of total 2 [TS]
{ id3_3_1_2e = +(a-b)*(c+c) -(a-c)*(b+c) +(a+c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 1 of total 2
{ id3_3_1_2f = +(a-b)*(a+b-c) -(a-c)*(a-b+c) -(b-c)*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the simplest three variable functional equation for the Weierstrass
\\ sigma function.
\\ Robert Fricke, Die Elliptischen Funktionen und ihre Anwendugen, Vol. 2,
\\ 1922, Teubner, Leipzig and Berlin, p. 159 "
\\ -\sigma(u+u_1)\sigma(u-u_1)\sigma(v)^2+\sigma(v+u_1)\sigma(v-u_1)\sigma(u)^2
\\ +\sigma(u+v)\sigma(u-v)\sigma(u_1)^2=0."
\\ A. O. L. Atkin and P. Swinnerton-Dyer, "Some Properties of partitions",
\\ Proc. London Math. Soc. (3) 4, (1954), pp. 84-106. p. 89 "Lemma 4. Suppose
\\ that none of b, c, d, b+-c, c+-d, b+-c is divisible by q. Then we have
\\ $$ P^2(b)P(c+d)P(c-d)-P^2(c)P(b+d)*P(b-d)+y^{c-d}P^2(d)P(b+c)P(b-c) = 0 $$
\\ Zhi-Guo Liu, "A three-term theta function identity and its applications",
\\ Advances in Mathematics, 195 (2005), pp. 1-23. p. 12. "Theorem 11. We have
\\ $$ \theta_1^2(z|\tau)\theta_1(x-y|\tau)\theta_1(x+y|\tau) =
\\ \theta_1^2(x|\tau)\theta_1(z-y|\tau)\theta_1(z+y|\tau) -
\\ \theta_1^2(y|\tau)\theta_1(z-x|\tau)\theta_1(z+x|\tau). (5.2) $$"
\\ {} 3 vars with 3 terms highest degree 1 of total 4 [WS]
{ id3_3_1_4a = +a*a*(b+c)*(b-c) -b*b*(a+c)*(a-c) +c*c*(a+b)*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {a -> (b+c)/2, b -> (-a+c)/2, c -> (b-a)/2} to get id3_3_1_4a.
\\ {} 3 vars with 3 terms highest degree 1 of total 4 [WS]
{ id3_3_1_4b = +(a+a)*(b+b-c-c)*(a-b-c)*(a-b-c)
-(b+b)*(a+a-c-c)*(a-b+c)*(a-b+c)
+(c+c)*(a+a-b-b)*(a+b-c)*(a+b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ R. W. Gosper and R. C. Schroeppel, "Somos Sequence Near-Addition Formulas
\\ and Modular Theta Functions", arXiv:math/0703470, p. 9. "With the relation
\\ A_{-n} = -A_n along with linear changes of variable, this can be rewritten
\\ $$ A_{-j}A_{j-k}A_{j-n}A_{n+k+j} + A_{2j}A_{-k}A_{-n}A_{n+k} = A_{-j}A_{k+j}
\\ A_{-n-k+j}A_{n+j}, $$ so that each term's subscript sum is 2j."
\\ Replace {j -> a, k -> b, n -> c} giving id3_3_1_4c.
\\ Zhi-Guo Liu, "A three-term theta function identity and its applications",
\\ Advances in Mathematics, 195 (2005), pp. 1-23. p. 7. "Theorem 8. We have
\\ $$ \theta_1(z+x|\tau)\theta_1(z+y|\tau)\theta_1(z-x-y|\tau)
\\ - \theta_1(z-x|\tau)\theta_1(z-y|\tau)\theta_1(z+x+y|\tau)
\\ = - \theta_1(x|\tau)\theta_1(y|\tau)\theta_1(x+y|\tau)
\\ \frac{\theta_1(2z|\tau),\theta_1(z|\tau)}. (3.9) $$"
\\ {} 3 vars with 3 terms highest degree 1 of total 4 [WS]
{ id3_3_1_4c = +(a+a)*b*c*(b+c)
+a*(a+b)*(a+c)*(a-b-c) -a*(a-b)*(a-c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 2 [NC]
{ id3_3_2_2a = +(a*a-b*c) -a*(a-c) -(a-b)*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 2
{ id3_3_2_2b = +(a*a-b*c) -b*(b-c) -(a-b)*(a+b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ If a(n) := q^n - 1, then a(2*n)*a(1) = a(n+1)*a(n) - q*a(n-1)*a(n). This
\\ is equivalent to (q^n+1)*(q-1) = (q^n*q - 1) - (q^n - q). Replace {q -> y,
\\ q^n -> x} to get (x+1)*(y-1) = (x*y-1) - (x-y). Replace {x -> b/a, y -> c/a}.
\\ See S. Ekhad, arXiv:1101.4060 for identity {2n} = {n+1}{n} + t{n-1}{n}.
\\ {} 3 vars with 3 terms highest degree 2 of total 2
{ id3_3_2_2c = +(a*a-b*c) +a*(b-c) -(a+b)*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From J. Michael Steele "ThreeLetterIdentities.pdf" as "EXAMPLE 2: NEWTON
\\ GIRARD $$ u^2+v^2+w^2 = (u+v+w)^2 - 2(uv+vw+uw) = e_1^2 - 2e_2 $$" n=3.
\\ {} 3 vars with 3 terms highest degree 2 of total 2 [CT]
{ id3_3_2_2d = +(a*a+b*b+c*c) -(a+b+c)*(a+b+c) +2*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is one formulation of Stewart's Theorem of a triangle with Cevian.
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3a = +a*(b*b-c*c) +b*(a*a-c*c) -(a+b)*(a*b-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3b = +(a+b)*(a*a+a*c-b*c) -a*a*(a+b+c) +b*b*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3c = +(a-b)*(a*b-c*c) -(a-c)*(a*c-b*b) -(a*a-b*c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 438). : "[[...]] 523.
\\ We collect here for reference a list of identities which are useful in the
\\ transformation of algebraical expressions; the student should verify these
\\ identities [[...]] $$ (a+b+c)(bc+ca+ab) -abc = (b+c)(c+a)(a+b). $$ [[...]]"
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3d = +a*b*c +(a+b)*(a+c)*(b+c) -(a+b+c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3e = +a*a*(b-c) -b*(a-c)*(a+c) +c*(a*a-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3f = +a*a*b -(a-c)*(a*b-c*c) -c*(a*b+a*c-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3g = +a*a*b -(a-c)*(a+c)*(b-c) -c*(a*a+b*c-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3h = +a*(a*b+a*c-b*b-c*c) +b*(a*b+b*c-a*a-c*c)
+c*(a*c+b*c-a*a-b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3i = +a*a*(b-c) -(a-b)*(a*b-a*c-b*c) -b*b*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3j = +a*a*a -(a+b)*(a*a-a*b-b*c) -b*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3k = +a*a*a -(a-b)*(a*a-b*c) -b*(a*a+a*c-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 3
{ id3_3_2_3l = +a*(a-c)*(a-c) -(a+b)*(a+b)*(a-b-c) +(b+c)*(a*a-a*b-a*c-b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 293) :
\\ "[[...]] E. Catalan attributed to J. Neuberg the identity $$ (a^2+b^2+c^2
\\ +bc+ca+ab)^2 = (a+b+c)^2(a^2+b^2+c^2) +(bc+ca+ab)^2. $$ [[...]]]".
\\ {} 3 vars with 3 terms highest degree 2 of total 4
{ id3_3_2_4a = +(a+b+c)*(a+b+c)*(a*a+b*b+c*c) +(a*b+b*c+a*c)*(a*b+b*c+a*c)
-(a*a+b*b+c*c+a*b+a*c+b*c)*(a*a+b*b+c*c+a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 4
{ id3_3_2_4b = +a*a*(b+c)*(b+c) +(a*a-b*c)*(a*a-b*c) -(a*a+b*b)*(a*a+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J. Cigler, "Sum of cubes: Old proofs suggest new q-analogues",
\\ arXiv:1403.6609, p. 4: "... which follows from $$ (1-q^{(n+1 2)})^2 -
\\ q^n(1-q^{(n 2)})^2 = ... = (1-q^n)(1-q^{n^2})." Replace {q^(n/2) -> b/a,
\\ q^(n^2/2) -> -c/a}.
\\ {} 3 vars with 3 terms highest degree 2 of total 4
{ id3_3_2_4c = +a*a*(b+c)*(b+c) -(a*a+b*c)*(a*a+b*c)
+(a-b)*(a+b)*(a-c)*(a+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 4
{ id3_3_2_4d = +(a*b+a*b)*(a*b+a*b) +(a+b+c)*(a+b-c)*(a-b+c)*(a-b-c)
-(a*a+b*b-c*c)*(a*a+b*b-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ A proof of Brahmagupta's formula uses this difference of two squares.
\\ {} 3 vars with 3 terms highest degree 2 of total 4
{ id3_3_2_4e = +a*b*(a+a)*(b+b) -(a*a+b*b-c*c)*(a*a+b*b-c*c)
+(a+b+c)*(a-b+c)*(a+b-c)*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 439, Examples, 23).
\\ {} 3 vars with 3 terms highest degree 2 of total 6
{ id3_3_2_6a = +a*b*c*(a+b+c)*(a+b+c)*(a+b+c)
-(a*b+a*c+b*c)*(a*b+a*c+b*c)*(a*b+a*c+b*c)
-(a*a-b*c)*(b*b-a*c)*(c*c-a*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 6 [CT]
{ id3_3_2_6b = +8*a*a*b*b*c*c +(a+b+c)*(a-b+c)*(a+b-c)*(a-b-c)*(a*a+b*b+c*c)
-(a*a+b*b-c*c)*(a*a-b*b+c*c)*(a*a-b*b-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 6
{ id3_3_2_6c = +(a-b)*(a-b)*(a+b)*(a+b)*c*c -(a-c)*(a-c)*(a+c)*(a+c)*b*b
+(a*a-b*c)*(a*a+b*c)*(b-c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 2 of total 9
{ id3_3_2_9 = +(a-b)*(a*b-c*c)*(a*b-c*c)*(a*b-c*c)*(a*a+a*b+b*b)
-(a-c)*(a*c-b*b)*(a*c-b*b)*(a*c-b*b)*(a*a+a*c+c*c)
-(b-c)*(a*a-b*c)*(a*a-b*c)*(a*a-b*c)*(b*b+b*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 3 of total 3 [NC]
{ id3_3_3_3a = +(a*a*a-b*c*c) -b*(a*a-c*c) -(a-b)*a*a ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 3 of total 3 [NC]
{ id3_3_3_3b = +(a*a*a-b*c*c) -a*(a*a-c*c) -(a-b)*c*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From J. Michael Steele "ThreeLetterIdentities.pdf" as "EXAMPLE 3: VIETA
\\ ARGUMENT AND CUBE SUM FACTORIZATION ... an amusing factorization:
\\ $$ u^3+v^3+w^3 - 3uvw = (u+v+w)(uv+vw+uv-u^2-v^2-w^2) $$ [[has 2 typos]]
\\ {} 3 vars with 3 terms highest degree 3 of total 3 [CT]
{ id3_3_3_3c = +(a*a*a+b*b*b+c*c*c) -3*a*b*c
+(a+b+c)*(a*b+a*c+b*c-a*a-b*b-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From "https://www.imo-official.org/problems/IMO2021SL.pdf" problem A8.
\\ {} 3 vars with 3 terms highest degree 3 of total 3 [CU]
{ id3_3_3_3d = +(a-b)*(b-c)*(c-a)
-(a*b*b+b*c*c+c*a*a) +(a*a*b+b*b*c+c*c*a) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J. Cigler, "Sum of cubes: Old proofs suggest new q-analogues",
\\ arXiv:1403.6609, p. 5: "A computational proof uses the fact that $ [n]_q^2
\\ - q[n-1]_q^2 = [2n-1]_q. $". Replace {q -> b/a, q^(n-1) -> c/a}.
\\ {} 3 vars with 3 terms highest degree 3 of total 4
{ id3_3_3_4 = +a*b*(a-c)*(a-c) +(a-b)*(a*a*a-b*c*c) -(a*a-b*c)*(a*a-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 3 of total 5
{ id3_3_3_5 = +a*a*a*(b-c)*(b-c) +(a+b)*(a+b)*c*(a*b-a*c+b*c)
-(a+c)*b*(a*a*b-a*a*c+a*b*c+b*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 3 of total 6 [CT]
{ id3_3_3_6 = +(a*a+a*b+b*b)*(a*a+a*c+c*c)*(b*b+b*c+c*c)
-(a-b)*(a-b)*(b-c)*(b-c)*(a-c)*(a-c)
-3*(a*a*b+a*c*c+b*b*c)*(a*a*c+a*b*b+b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 4 of total 4
{ id3_3_4_4a = +(a*a*a*a-b*b*c*c) -a*(a-c)*(a*a+b*c) -c*(a-b)*(a*a+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 4 of total 4 [CT]
{ id3_3_4_4b = +2*(a*a*a*a+b*b*b*b+c*c*c*c) -(a*a+b*b+c*c)*(a*a+b*b+c*c)
-(a+b+c)*(a+b-c)*(a-b+c)*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 4 of total 6
{ id3_3_4_6 = +a*a*a*(a-b)*(a*a+a*b+b*b) +(a*b-c*c)*(a*a*b*b+a*b*c*c+c*c*c*c)
-(a-c)*(a+c)*(a*a-a*c+c*c)*(a*a+a*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 4 of total 7
{ id3_3_4_7 = +(a*a*a-b*c*c)*(a*b*b*b+c*c*c*c) +c*c*(a*b-c*c)*(a*a*a+b*b*b)
-b*(a-c)*(a+c)*(a*a+c*c)*(b*b+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 4 of total 8
{ id3_3_4_8 =
+a*a*b*b*c*c*(a+b+c)*(a+b+c)
+(a*b+a*c+b*c)*(a*b+a*c+b*c)*(a*a*b*b+a*a*c*c+b*b*c*c)
-(a*a*b*b+a*a*b*c+a*b*b*c+a*a*c*c+a*b*c*c+b*b*c*c)*(a*a*b*b+a*a*b*c+a*b*b*c+a*a*c*c+a*b*c*c+b*b*c*c)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 4 of total 9
{ id3_3_4_9 = +(a-b)*(a*b-c*c)*(a*a+a*b+b*b)*(a*a*b*b+a*b*c*c+c*c*c*c)
-(a-c)*(a*c-b*b)*(a*a+a*c+c*c)*(a*a*c*c+a*b*b*c+b*b*b*b)
-(b-c)*(a*a-b*c)*(b*b+b*c+c*c)*(a*a*a*a+a*a*b*c+b*b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 3 terms highest degree 4 of total 12
{ id3_3_4_12 = +(a-b)*(a*a+a*b+b*b)*(a*a*a-b*c*c)*(a*a*a-b*c*c)*(a*a*a-b*c*c)
-(a-c)*(a*a+a*c+c*c)*(a*a*a-b*b*c)*(a*a*a-b*b*c)*(a*a*a-b*b*c)
+a*a*a*(b-c)*(b-c)*(b-c)*(a*a-b*c)*(a*a*a*a+a*a*b*c+b*b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 1 [NC]
{ id3_4_1_1 = +(a-b) -(a+b) +(b-c) +(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a rare identity notable for having two tags.
\\ {} 3 vars with 4 terms highest degree 1 of total 2 [NC,TS]
{ id3_4_1_2a = +a*(b+c) -a*(b-c) -(a+b)*c -(a-b)*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id3_3_2_2a with its first term split into two terms.
\\ Equivalent to (c2-c3)*c3 - c4*c2 = (c2-c3-c4)*c2 - (c2-c3)*(c2-c3).
\\ {} 3 vars with 4 terms highest degree 1 of total 2 [NC]
{ id3_4_1_2b = +a*a -b*c -a*(a-c) -(a-b)*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 2
{ id3_4_1_2c = +a*a -b*c -b*(b-c) -(a-b)*(a+b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 2
{ id3_4_1_2d = +a*a -b*c +a*(b-c) -(a+b)*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 2 [NC]
{ id3_4_1_2e = +(a-b)*(a-b) +(a-b)*(b-c) -(a-c)*(a-c) +(b-c)*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 2 [NC]
{ id3_4_1_2f = +a*b -b*c +(a+b)*c -a*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 2 [TS]
{ id3_4_1_2g = +a*(a-b+c) -a*(a+b-c) +(a+b)*(a-c) -(a-b)*(a+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 2 [TS]
{ id3_4_1_2h = +a*c +(a+b)*(b-c) +(a-b-b)*(a+a-c) +(a-b)*(b+c-a-a) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 2
{ id3_4_1_2i = +(a+a)*(b+b) -(a+b)*(a+b+c) +a*(a-b+c) +b*(b-a+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ A three variable functional equation for Jacobi elliptic functions.
\\ This is one expansion of the 3x3 Vandermonde determinant.
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957, Chap. XXXIV, Art. 523,
\\ (p. 438). : "$$ \Sigma bc(b-c) = -(b-c)(c-a)(a-b). $$"
\\ G. Chang and T. W. Sederberg, "Over and Over Again", MAA, 1997. (p. 72):
\\ solution to Problem 13.2. "The identity $$ z_1z_2(z_1-z_2)+z_2z_3(z_2-z_3)+
\\ z_3z_1(z_3-z_1)=(z_1-z_2)(z_2-z_3)(z_3-z_1) $$ can be easily verified."
\\ {} 3 vars with 4 terms highest degree 1 of total 3 [JE]
{ id3_4_1_3a = +a*b*(a-b) +b*c*(b-c) -a*c*(a-c) -(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ T. Miwa, "On Hirota's Difference Equations", Proc. Japan Acad.,
\\ 58, Ser. A (1982), pp. 9-12: "[...] Theorem 4. [...] solves
\\ the following bilinear difference equation. (4.7) (a+b)(a+c)(b-c)
\\ [...]+(b+c)(b+a)(c-a)[...]+(c+a)(c+a)(a-b)[...]+(a-b)(b-c)(c-a)[...]"
\\ If denominators are nonzero, then -(a-b)(b-c)(c-a)/((a+b)(b+c)(c+a)) =
\\ (a-b)/(a+b) + (b-c)/(b+c) + (c-a)/(c+a).
\\ Replace {a -> b+c, b -> a+c, c -> a+b} in id3_4_1_3a.
\\ {} 3 vars with 4 terms highest degree 1 of total 3 [JE]
{ id3_4_1_3b = +(a+b)*(a+c)*(b-c) -(a+b)*(a-c)*(b+c)
+(a-b)*(a+c)*(b+c) -(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is one expansion of the 3x3 Vandermonde determinant. For example in
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 420, Examples, 15).
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 438). : "should verify
\\ these identities. ... $$ \Sigma a^2(b-c) = -(b-c)(c-a)(a-b). $$"
\\ {} 3 vars with 4 terms highest degree 1 of total 3
{ id3_4_1_3c = +a*a*(b-c) -b*b*(a-c) +c*c*(a-b) -(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id3_4_1_3c multiplied by 2 with a nontrivial functional equation.
\\ {} 3 vars with 4 terms highest degree 1 of total 3 [TS,CT]
{ id3_4_1_3d = +a*a*(b+b-c-c) -b*b*(a+a-c-c) +c*c*(a+a-b-b)
-2*(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is one expansion of the 3x3 Vandermonde determinant.
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 438). : "should verify
\\ these identities. ... $$ \Sigma a(b^2-c^2) = (b-c)(c-a)(a-b). $$"
\\ C. G. J. Jacobi, "Formulae Novae in Theoria Transcendentium Ellipticarum
\\ Fundamentales", in Werke, Bd. I, (1881) 333-341, p. 338.:
\\ " (t'^2-t''^2)t +(t''^2-t^2)t' +(t^2-t'^2)t'' = (t'-t'')(t''-t)(t-t')."
\\ {} 3 vars with 4 terms highest degree 1 of total 3
{ id3_4_1_3e = +a*(b-c)*(b+c) -b*(a-c)*(a+c)
+c*(a-b)*(a+b) +(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id3_4_1_3e multiplied by 2 with a nontrivial functional equation.
\\ {} 3 vars with 4 terms highest degree 1 of total 3 [TS,CT]
{ id3_4_1_3f = +(a+a)*(b-c)*(b+c) -(b+b)*(a-c)*(a+c)
+(c+c)*(a-b)*(a+b) +2*(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This may be simplest functional equation for Jacobi elliptic functions.
\\ Replace {b -> a+b+c, c -> a+c} in id3_4_1_3a.
\\ {} 3 vars with 4 terms highest degree 1 of total 3 [JE]
{ id3_4_1_3g = +a*(b+c)*(a+b+c) -a*c*(a+c) -b*(a+c)*(a+b+c) +b*c*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 3
{ id3_4_1_3h = +a*a*b -a*a*(b-c) -c*(a-b)*(a+b) -b*b*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 3
{ id3_4_1_3i = +a*b*(a-b) -c*(a-b)*(a+b) +(a-b)*c*c -(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 3 [TS]
{ id3_4_1_3j = +a*(b+c)*(a+b+c) -a*b*(a+b+c+c) -(a+b)*(a+c)*(b+c)
+b*(a+c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, New First Course in the Theory of Equations, Wiley,
\\ 1939. (p. 122, Prob. 10) [variant 3x3 Vandermonde determinant].
\\ {} 3 vars with 4 terms highest degree 1 of total 4
{ id3_4_1_4a = +a*b*(a-b)*(a+b) -a*c*(a-c)*(a+c) +b*c*(b-c)*(b+c)
-(a-b)*(a-c)*(b-c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id3_4_1_a multiplied by 2 with a nontrivial functional equation.
\\ {} 3 vars with 4 terms highest degree 1 of total 4 [TS]
{ id3_4_1_4b = +a*b*(a-b)*(a+a+b+b) -a*c*(a-c)*(a+a+c+c) +b*c*(b-c)*(b+b+c+c)
-(a-b)*(a-c)*(b-c)*(a+a+b+b+c+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 439, Examples, 16).
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 549) :
\\ "P. G. Tait^{23} noted that x^3 + y^3 = z^3 implies $$
\\ (x^3+z^3)^3y^3 + (x^3-y^3)z^3 = (x^3+y^3)x^3 $$ ..."
\\ {} 3 vars with 4 terms highest degree 1 of total 4
{ id3_4_1_4c = +a*(b-c)*(b-c)*(b-c) -b*(a-c)*(a-c)*(a-c) +c*(a-b)*(a-b)*(a-b)
+(a-b)*(a-c)*(b-c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id3_4_1_4c multiplied by 3 with a nontrivial functional equation.
\\ {} 3 vars with 4 terms highest degree 1 of total 4 [TS,CT]
{ id3_4_1_4d = -(a+a+a)*(b-c)*(b-c)*(b-c) +(b+b+b)*(a-c)*(a-c)*(a-c)
-(c+c+c)*(a-b)*(a-b)*(a-b) -3*(a-b)*(a-c)*(b-c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 4 [TS]
{ id3_4_1_4e = +a*(a-b)*(a+b)*(b-c) +a*(b-c)*(b-c)*(b+c)
-b*(a-c)*(a-c)*(a+c) +c*(a-b)*(a-c)*(a+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Similar to id3_4_1_4c and id3_4_1_4d but better.
\\ "mathlover", "https://math.stackexchange.com/q/2395234/", Aug 16 2017
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 549) :
\\ "P. G. Tait^{23} noted that x^3 + y^3 = z^3 implies $$
\\ (x^3+z^3)^3y^3 + (x^3-y^3)z^3 = (x^3+y^3)x^3 $$ ..." Replace {x^3 -> a,
\\ y^3 -> b, z^3 -> -c}.
\\ {} 3 vars with 4 terms highest degree 1 of total 4 [TS]
{ id3_4_1_4f = +a*a*a*(b-c) -b*b*b*(a-c) +c*c*c*(a-b)
-(a-b)*(a-c)*(b-c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 4 [WS]
{ id3_4_1_4g = +(a+b)*(a+c)*(a-b-c)*(b-c) -(a-b)*(a+c)*(a+b-c)*(b+c)
-(a-b)*(a-c)*(a+b+c)*(b-c) +(a+b)*(a-c)*(a-b+c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 4
{ id3_4_1_4h = +a*a*(b-c)*(b-c) -a*b*(a-c)*(b-c) +b*c*(a-b)*(a-c)
-c*c*(a-b)*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 1 of total 4
{ id3_4_1_4i = +a*(a+b+b)*(a+b+b)*(a+b+b) -b*(a+a+b)*(a+a+b)*(a+a+b)
+(a-b)*(a-b)*(a-b)*c -(a-b)*(a-b)*(a-b)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. E. Andrews and D. Zeilberger, "A Short Proof of Jacobi's Formula for
\\ the number of representations of an integer as a sum of four squares",
\\ arXiv:math/9206203v1. Equation (1) 2nd part is equivalent to:
\\ q^k(1-q^{n+1}) - (1+q^{n+1}) = - (1+q^k)q^{n+1} - (1-q^k).
\\ Replace {q^{n+1} -> c/a, q^k -> b/a}.
\\ {} 3 vars with 4 terms highest degree 2 of total 3
{ id3_4_2_3a = +a*a*(a-b) -a*(a*a+c*c) +b*(a-c)*(a+c) +c*c*(a+b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 3
{ id3_4_2_3b = +a*b*b -a*(a+b)*(b-c) +b*(a+a-b)*(a-c-c)
-(a-b)*(a*b+a*c-b*c-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 3
{ id3_4_2_3c = +a*(a*a-b*c) -b*(a*c-b*b) -c*(a*b-c*c)
-(a+b+c)*(a*a-a*b-a*c+b*b-b*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 266) :
\\ "Catalan gave the identity $$ (a^2+b^2+c^2+ab+bc+ac)^2 = (a+c)^2(a+b)^2 +
\\ (b+c)^2(a+b)^2+(c^2+ac+bc-ab)^2 $$".
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4a = +(a+b)*(a+b)*(a+c)*(a+c) +(a+b)*(a+b)*(b+c)*(b+c)
+(a*b-a*c-b*c-c*c)*(a*b-a*c-b*c-c*c)
-(a*a+b*b+c*c+a*b+a*c+b*c)*(a*a+b*b+c*c+a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4b = +(a*b+a*c+b*c)*(a*b+a*c+b*c) +b*b*(a+b+c)*(a+b+c)
-b*(a+b+c)*(a*b+a*c+b*c) -(a*a+a*b+b*b)*(b*b+b*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4c = +c*c*(a-b)*(a+b) -b*(a+c)*(a*c-b*b) -b*b*(b-c)*(b+c)
+(a-b)*(b-c)*(a*c-b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4d = +a*b*b*(a-c) +a*c*(b-c)*(b-c) -c*c*c*(a-c)
-(a*b-c*c)*(a*b-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4e = +a*(a*a+a*b+b*c)*(b-c) -a*(a+b+c)*(a*a-b*c)
+(a*a+a*c+b*c)*(a*a+a*c+b*c) -(a*b+a*c+b*c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4 [CT]
{ id3_4_2_4f = +(a-b)*(a-b)*(a-b)*(a-b) +(a-c)*(a-c)*(a-c)*(a-c)
+(b-c)*(b-c)*(b-c)*(b-c)
-2*(a*a+b*b+c*c-a*b-a*c-b*c)*(a*a+b*b+c*c-a*b-a*c-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4g = +a*a*b*(a+b) -a*b*(a-c)*(a+c) -(a-b)*b*c*c -b*b*(a*a+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4h = +a*a*(a+b+c)*(a+b+c) +a*(b-c)*(a*a+a*c+b*c)
-(a*a-b*c)*(a*b+a*c+b*c) -(a*a+a*b+b*c)*(a*a+a*b+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4i = +a*(b-c)*(b*b+b*c+c*c) -b*(a-c)*(a*a+a*c+c*c)
+c*(a-b)*(a*a+a*b+b*b) +(a-b)*(a-c)*(b-c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4j = +(a*b+a*b)*(a*b+a*b) +(a*c+a*c)*(a*c+a*c)
-(a*a+b*b+c*c)*(a*a+b*b+c*c) +(a*a-b*b-c*c)*(a*a-b*b-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 4
{ id3_4_2_4k = +(a+b)*(a+b)*(a+b)*(a+b) -(a+a)*(b+b)*(a+b)*(a+b)
-(a*a-b*b-c*c)*(a*a-b*b+c*c) -c*c*c*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 5
{ id3_4_2_5a = +a*a*a*a*(b-c) +b*b*b*b*(c-a) +c*c*c*c*(a-b)
+(a-b)*(b-c)*(c-a)*(a*a+a*b+a*c+b*b+b*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 5
{ id3_4_2_5b = +a*a*b*b*(a-b) -a*a*c*c*(a-c) +b*b*c*c*(b-c)
-(a-b)*(a-c)*(b-c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 5
{ id3_4_2_5c = +a*a*a*(b-c)*(b+c) -b*b*b*(a-c)*(a+c) +c*c*c*(a-b)*(a+b)
-(a-b)*(a-c)*(b-c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 5
{ id3_4_2_5d = +a*a*(b-c)*(b-c)*(b-c) -b*b*(a-c)*(a-c)*(a-c)
+c*c*(a-b)*(a-b)*(a-b) +(a-b)*(a-c)*(b-c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 2 of total 5
{ id3_4_2_5e = +(a+b)*(a+c)*(b-c)*(a*b+a*c-b*c) -(a-b)*(a+c)*(b+c)*(a*b-a*c-b*c)
+(a-b)*(a-c)*(b-c)*(a*b+a*c+b*c) -(a+b)*(a-c)*(b+c)*(a*b-a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From J. Michael Steele "ThreeLetterIdentities.pdf" as "EXAMPLE 2: NEWTON
\\ GIRARD $$ u^3+v^3+w^3 = (u+v+w)^3 - 3(u+v+w)(uv+vw+uw) + 3uvw = e_1^3 -
\\ 3e_1e_2 + 3e_3 $$" n=4.
\\ {} 3 vars with 4 terms highest degree 3 of total 3 [CT]
{ id3_4_3_3 = +(a*a*a+b*b*b+c*c*c) -(a+b+c)*(a+b+c)*(a+b+c)
+3*(a+b+c)*(a*b+a*c+b*c) -3*a*b*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, New First Course in the Theory of Equations, Wiley,
\\ 1939. (p. 122, Problem 11) [variant 3x3 Vandermonde determinant].
\\ {} 3 vars with 4 terms highest degree 3 of total 5
{ id3_4_3_5 = +a*a*(b*b*b-c*c*c) +b*b*(c*c*c-a*a*a) +c*c*(a*a*a-b*b*b)
+(a-b)*(a-c)*(b-c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 3 of total 6
{ id3_4_3_6a = +a*a*(a*a-b*c)*(a*a+b*c) -a*b*(a+c)*(a*a*a-b*b*c)
+b*b*c*c*(a-b)*(a+b) -(a-b)*(a*a-b*c)*(a*a*a-b*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 3 of total 6
{ id3_4_3_6b = +a*a*a*a*(a-c)*(a+c) -a*c*(a+b)*(a*a*a-b*c*c)
+c*c*(a*a-b*c)*(a*a+b*c) -(a-c)*(a*a-b*c)*(a*a*a-b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 596):
\\ "C. Hermite noted the solution $ x=a(ab-c^2), y=a^3-b^2c, z=b(c^2-ab),
\\ u=a^2c-b^3 $ of $$ x^2y+y^2z+z^u+u^2x = 0. $$
\\ {} 3 vars with 4 terms highest degree 3 of total 9
{ id3_4_3_9 = +a*a*(a*a*a-b*b*c)*(a*b-c*c)*(a*b-c*c)
+a*(a*b-c*c)*(a*a*c-b*b*b)*(a*a*c-b*b*b)
-b*(a*a*a-b*b*c)*(a*a*a-b*b*c)*(a*b-c*c)
+b*b*(a*a*c-b*b*b)*(a*b-c*c)*(a*b-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 3 of total 10
{ id3_4_3_10 = +a*a*a*(a*b-c*c)*(a*b-c*c)*(a*a*c-b*b*b)
+a*c*(a*b-c*c)*(a*a*c-b*b*b)*(a*a*c-b*b*b)
+b*b*b*(a*b-c*c)*(a*b-c*c)*(a*a*a-b*b*c)
-b*c*(a*b-c*c)*(a*a*a-b*b*c)*(a*a*a-b*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ L. G. Weld, Determinants. (p. 21) : "Prob. 17. That that the determinant
\\ |ab c^2 c^2 \\ a^2 bc a^2 \\ b^2 b^2 ca | contains the factor (bc+ca+ab)."
\\ {} 3 vars with 4 terms highest degree 4 of total 6
{ id3_4_4_6 = +a*a*b*b*(a*b-c*c) +a*a*c*c*(a*c-b*b) -b*b*c*c*(a*a-b*c)
-(a*b+a*c+b*c)*(a*a*b*b-a*a*b*c-a*b*b*c+a*a*c*c-a*b*c*c+b*b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 4 terms highest degree 4 of total 7
{ id3_4_4_7 = +a*a*a*b*b*b*(a-b) -a*a*a*c*c*c*(a-c) +b*b*b*c*c*c*(b-c)
-(a-b)*(a-c)*(b-c)*(a*a*b*b+a*a*b*c+a*b*b*c+a*a*c*c+a*b*c*c+b*b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. E. Andrews and D. Zeilberger, "A Short Proof of Jacobi's Formula for
\\ the number of representations of an integer as a sum of four squares",
\\ arXiv:math/9206203v1. Equation (1) 1st part is equivalent to:
\\ q^k(1-q^{n+1})(1+q^{n+1})^3(1+q^{n+k+1})(q^k+q^{n+1})
\\ -q^k(1+q^{n+1})(1-q^{n+1})^3(1-q^{n+k+1})(q^k-q^{n+1}) =
\\ q^{n+1}(1+q^k)^2(1+q^{2n+2})(q^k-q^{n+1})(1+q^{n+k+1})
\\ -q^{n+1}(1+q^k)^2(1+q^{2n+2})(q^k+q^{n+1})(1-q^{n+k+1}).
\\ Replace {q^{n+1} -> (b/a)^2, q^k -> -c/a}.
\\ {} 3 vars with 4 terms highest degree 4 of total 14
{ id3_4_4_14 =
+a*b*b*(a-c)*(a-c)*(a*c-b*b)*(a*a*a+b*b*c)*(a*a*a*a+b*b*b*b)
+a*b*b*(a-c)*(a-c)*(a*c+b*b)*(a*a*a-b*b*c)*(a*a*a*a+b*b*b*b)
-c*(a-b)*(a-b)*(a-b)*(a+b)*(a+b)*(a+b)*(a*a+b*b)*(a*c+b*b)*(a*a*a+b*b*c)
+c*(a-b)*(a+b)*(a*a+b*b)*(a*a+b*b)*(a*a+b*b)*(a*c-b*b)*(a*a*a-b*b*c)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Noam Elkies, "On A^4+B^4+C^4=D^4", Math. Comp., v. 51, Oct 1988, p. 826,
\\ "Bremner wrote (1) as $$ 2(1+r^2)(1+s^2) = (1+r^2+s^2)^2 + t^2 $$ ...".
\\ {} 3 vars with 5 terms highest degree 1 of total 2 [CT]
{ id3_5_1_2 = +a*a -b*b -c*c -2*(a+b)*(a+c) +(a+b+c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 17) :
\\ "(ii) If $a,b,c$ are the sides of a triangle then the case $n=3$ gives
\\ $abc \ge (a+b-c)(b+c-a)*(c+a-b).$ This is also known as Padoa's inequality."
\\ {} 3 vars with 5 terms highest degree 1 of total 3 [TS]
{ id3_5_1_3a = +a*b*c -a*(a-b)*(a-c) +(a-b)*b*(b-c) -(a-c)*(b-c)*c
+(a+b-c)*(a-b-c)*(a-b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ If x^3+y^3=z^3, then (x+y-z)^3 = 3(x+y)(z-x)(z-y). Related to FLT N=3.
\\ {} 3 vars with 5 terms highest degree 1 of total 3 [CT]
{ id3_5_1_3b = +a*a*a +b*b*b +c*c*c -(a+b+c)*(a+b+c)*(a+b+c)
+3*(a+b)*(a+c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From an answer to "https://math.stackexchange.com/q/3640206/" identity:
\\ "(1+x/y)(1+y/z)(1+z/x) = 2 + (x/y+y/z+z/x) + (y/x+z/y+x/z)."
\\ {} 3 vars with 5 terms highest degree 1 of total 3 [CT]
{ id3_5_1_3c = +2*a*b*c +a*b*(a+b) +a*c*(a+c) +b*c*(b+c)
-(a+b)*(a+c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 1 of total 3 [CT]
{ id3_5_1_3d = +2*a*b*c +a*a*(b+c) +b*b*(a+c) +c*c*(a+b)
-(a+b)*(a+c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 1 of total 3
{ id3_5_1_3e = +(a+a)*(b+b)*(c+c) -(a+b)*(a+c)*(b+c) +a*(b-c)*(b-c)
+b*(a-c)*(a-c) +c*(a-b)*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Chang and T. W. Sederberg, "Over and Over Again", MAA, 1997. (p. 15)
\\ "Problem 3.2 [IMO 1983/6]. Let $a,b$, and $c$ be the lengths of the sides
\\ of a triangle. Prove that $$ (3.7) a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0. $$"
\\ (p. 16) "Leeb pointed out that (3.7) has an equivalent form $$ (3.10)
\\ a(b-c)^2(b+c-a)+b(a-b)(a-c)(a+b-c)\ge 0. $$"
\\ {} 3 vars with 5 terms highest degree 1 of total 4
{ id3_5_1_4 = +a*a*b*(a-b) +b*b*c*(b-c) -a*c*c*(a-c) -a*(b-c)*(b-c)*(b+c-a)
-b*(a-b)*(a-c)*(a+b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 2 of total 2 [CT]
{ id3_5_2_2 = +2*(a*a+b*b+c*c) -2*(a*b+a*c+b*c) -(a-b)*(a-b) -(a-c)*(a-c)
-(b-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 2 of total 3 [CT]
{ id3_5_2_3a = +2*a*b*c -(a+b-c)*(a-b+c)*(a-b-c) +a*(a*a-b*b-c*c)
+b*(b*b-a*a-c*c) +c*(c*c-a*a-b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Sometimes referred to as "Gauss identity". For example in 'bluemaster',
\\ "https://math.stackexchange.com/q/2956878".
\\ {} 3 vars with 5 terms highest degree 2 of total 3 [CT]
{ id3_5_2_3b = +3*a*b*c +(a+b+c)*(a*a+b*b+c*c-a*b-a*c-b*c)
-a*a*a -b*b*b -c*c*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Benjamin Warren, "https://mathoverflow.net/q/443579/" second identity:
\\ $ u^3+(u+r)^3+v^3+(v+r)^3=(u+v+r)(2u^2-2uv+2v^2+ru+rv+2r^2). $
\\ {} 3 vars with 5 terms highest degree 2 of total 3
{ id3_5_2_3c = +a*a*a +(a+c)*(a+c)*(a+c) +b*b*b +(b+c)*(b+c)*(b+c)
-(a+b+c)*(a*a+a*a+b*b+b*b+c*c+c*c-a*b-a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 2 of total 3 [CT]
{ id3_5_2_3d = +9*a*b*c -(a+b+c)*(a*b+a*c+b*c) +a*(b-c)*(b-c)
+b*(a-c)*(a-c) +c*(a-b)*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 658):
\\ "E. Fauquembergue noted that, if $ \alpha^2+\beta^2=\gamma^2, $
\\ $ (\alpha\beta)^4 + (\beta\gamma)^4 + (\gamma\alpha)^4 = (\alpha^4
\\ + \alpha^2\beta^2 + \beta^4)^2, ...". Replace {\alpha^2 -> a, \beta^2 -> b,
\\ \gamma^2 -> c}.
\\ {} 3 vars with 5 terms highest degree 2 of total 4
{ id3_5_2_4a = +a*a*b*b +a*a*c*c +b*b*c*c -(a*a+a*b+b*b)*(a*a+a*b+b*b)
+(a+b+c)*(a+b-c)*(a*a+b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The Klein quartic a^3b +b^3c +c^3a can be written as Q_0Q_2 - Q_1Q_1 where
\\ Q_i are quadratic in a,b,c. One way is from an answer by Peter Mueller to
\\ MathOverflow question "https://mathoverflow.net/q/445951/".
\\ {} 3 vars with 5 terms highest degree 2 of total 4
{ id3_5_2_4b = +a*a*a*b +a*c*c*c +b*b*b*c +(a*a-a*b+b*c)*(a*a-a*b+b*c)
-(a*a-a*b+b*b+a*c)*(a*a-a*c+b*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id3_3_4_4b with its first term split into three terms.
\\ {} 3 vars with 5 terms highest degree 2 of total 4
{ id3_5_2_4c = +(a+a)*a*a*a +(b+b)*b*b*b +(c+c)*c*c*c
-(a*a+b*b+c*c)*(a*a+b*b+c*c)
-(a+b+c)*(a+b-c)*(a-b+c)*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 259) :
\\ "If $ a,b,c,x,y,z>0 $ where $ x=b+c-a, y=c+a-b, z=a+b-c $, then
\\ $$ 1/a+1/b+1/c \le \frac{1}/{xyz} $$".
\\ {} 3 vars with 5 terms highest degree 2 of total 5
{ id3_5_2_5a = -a*b*c*(a*a+b*b+c*c) -(a-b-c)*(a+b-c)*(a-b+c)*(a*b+a*c+b*c)
+a*b*(a+b)*(a-b)*(a-b) +a*c*(a+c)*(a-c)*(a-c)
+b*c*(b+c)*(b-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 2 of total 5
{ id3_5_2_5b = +a*a*a*c*(b-c) -a*b*b*b*(a-c) +a*(b-c)*(b-c)*(a*b+a*c-b*c)
+b*c*c*c*(a-b) -c*(a-b)*(a-c)*(a*b-a*c-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 2 of total 6 [CT]
{ id3_5_2_6a = +a*a*a*b*b*b +a*a*a*c*c*c +b*b*b*c*c*c
-(a*b+a*c+b*c)*(a*b+a*c+b*c)*(a*b+a*c+b*c)
+3*a*b*c*(a+b)*(a+c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 2 of total 6
{ id3_5_2_6b = +a*a*b*b*c*c -a*a*b*b*(a-c)*(b-c) +a*a*c*c*(a-b)*(b-c)
-b*b*c*c*(a-b)*(a-c) +(a*b-a*c-b*c)*(a*b+a*c-b*c)*(a*b-a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 2 of total 6
{ id3_5_2_6c = +a*a*b*b*c*c +a*a*a*a*b*b +a*a*b*b*b*b
-c*c*(a*a+a*b+b*b)*(a*a+a*b+b*b)
-(a*a+b*b)*(a*b-a*c-b*c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From "https://math.stackexchange.com/q/3725797" answer by "NKellira".
\\ {} 3 vars with 5 terms highest degree 2 of total 6
{ id3_5_2_6d = +a*b*c*(a+b)*(b+c)*(c+a) +a*c*(a-b)*(a-b)*(b*b+c*c)
+a*b*(b-c)*(b-c)*(a*a+c*c) +b*c*(a-c)*(a-c)*(a*a+b*b)
-(a*a+b*c)*(b*b+c*a)*(c*c+a*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 439, Examples, 23).
\\ {} 3 vars with 5 terms highest degree 3 of total 6
{ id3_5_3_6 = +a*a*a*b*b*b +a*a*a*c*c*c +b*b*b*c*c*c
+(a*a-b*c)*(b*b-a*c)*(c*c-a*b) -a*b*c*(a*a*a+b*b*b+c*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 4 of total 6 [CT]
{ id3_5_4_6 = +2*a*a*b*b*c*c +a*b*(a*a*b*b-a*a*c*c-b*b*c*c)
-a*c*(a*a*b*b-a*a*c*c+b*b*c*c) -b*c*(a*a*b*b+a*a*c*c-b*b*c*c)
-(a*b-a*c-b*c)*(a*b+a*c-b*c)*(a*b-a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 4 of total 7
{ id3_5_4_7a = +a*a*a*a*(a-b)*(a-b)*(a+b)
+a*a*a*a*(a-c)*(a*a-b*c)
+b*c*(a-b)*(a+b)*(a+b)*(a*a-b*c)
-a*(a-b)*(a-b)*(a-c)*(a+b)*(a*a-b*c)
-(a*a*a-b*b*c)*(a*a*a*a-b*b*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 4 of total 7
{ id3_5_4_7b = +a*a*a*a*(a-b)*(a*a+b*c)
+a*a*a*a*(a+b)*(a*a-b*c)
-a*(a-b)*(a+b)*(a*a+b*c)*(a*a-b*c)
-b*b*(a-c)*(a*a-b*c)*(a*a+b*c)
-(a*a*a-b*b*c)*(a*a*a*a+b*b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 4 of total 7
{ id3_5_4_7c = +a*a*a*a*(b-c)*(b*b-a*c)
+a*a*c*c*(a-b)*(a-b)*(a+b)
-a*(a-b)*(a-b)*(a+b)*(b-c)*(b*b-a*c)
-b*b*(a-b)*(a+b)*(a+b)*(b*b-a*c)
-(b*b*b-a*a*c)*(b*b*b*b-a*a*a*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 5 terms highest degree 4 of total 7
{ id3_5_4_7d = +a*a*a*a*(b-c)*(b-c)*(b+c) -a*b*c*(a*a*b*b+a*a*c*c+b*b*c*c)
+b*b*b*b*(a-c)*(a-c)*(a+c) +c*c*c*c*(a-b)*(a-b)*(a+b)
-(a+b+c)*(a*b-a*c-b*c)*(a*b+a*c-b*c)*(a*b-a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From solution of x^4+y^4+z^4=t^4 using x=ab, y=ac, z=bc where a^2+b^2=c^2.
\\ {} 3 vars with 5 terms highest degree 4 of total 8
{ id3_5_4_8 = +a*a*a*a*b*b*b*b +a*a*a*a*c*c*c*c +b*b*b*b*c*c*c*c
+(a*a*a*a+b*b*b*b)*(a*a+b*b-c*c)*(a*a+b*b+c*c)
-(a*a-a*b+b*b)*(a*a-a*b+b*b)*(a*a+a*b+b*b)*(a*a+a*b+b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 260):
\\ "[[...]] , Goldbach gave the identity $$ \beta^2+\gamma^2+(3\delta-\beta
\\ -gamma)^2=(2\delta-\beta)^2+(2\delta-\gamma)^2+(\delta-\beta-\gamma)^2. $$"
\\ {} 3 vars with 6 terms highest degree 1 of total 2
{ id3_6_1_2a = +(a-b)*(a-b) +(a-c)*(a-c) +(a+b+c)*(a+b+c)
-(a+b)*(a+b) -(a+c)*(a+c) -(a-b-c)*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ N. J. Wildberger, Divine Proportions, 2005, (p. 69, Exer. 5.14,5.15) :
\\ "Exercise 5.14 Show that $$ A(Q_1,Q_2,Q_3) = Q_1(Q_2+Q_3-Q_1) +Q_2
\\ (Q_3+Q_1-Q_2) +Q_3(Q_1+Q_2-Q_3). $$
\\ Exercise 5.15 Show that $$ A(Q_1,Q_2,Q_3) = (Q_3+Q_1-Q_2)(Q_1+Q_2-Q_3)
\\ +(Q_1+Q_2-Q_3)(Q_2+Q_3-Q_1) +(Q_2+Q_3-Q_1)(Q_3+Q_1-Q_2). $$"
\\ {} 3 vars with 6 terms highest degree 1 of total 2
{ id3_6_1_2b = -a*(a-b-c) +b*(a-b+c) +c*(a+b-c) -(a-b+c)*(a+b-c)
+(a+b-c)*(a-b-c) +(a-b-c)*(a-b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 6 terms highest degree 1 of total 3 [JE]
{ id3_6_1_3 = +a*b*(a-b) -a*c*(a-c) +b*c*(b-c) -(a+b)*(a+c)*(b-c)
+(a+b)*(a-c)*(b+c) -(a-b)*(a+c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Derived from identity s(a+b)s(a-b) = (s(a)^2-s(b)^2)/(1-(ks(a)s(b))^2).
\\ {} 3 vars with 6 terms highest degree 1 of total 6 [JE,ZS]
{ id3_6_1_6 = +a*a*b*b*(a-b)*(a+b) -b*b*c*c*(a-b)*(a+b)
-b*b*(a-b)*(a+b)*(a-c)*(a+c) -a*a*c*c*(a-c)*(a+c)
+b*b*c*c*(a-c)*(a+c) +c*c*(a-b)*(a+b)*(a-c)*(a+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Z. Daroczy, "Ramanujan's identities and functional equations", Aequationes
\\ Math. 58 (1999), 41-45. p. 42, equation (10).
\\ {} 3 vars with 6 terms highest degree 2 of total 4
{ id3_6_2_4a = +a*a*(a+b+c)*(a+b+c) -a*a*(b-c)*(b-c)
+a*(a*a+a*b+b*c)*(b-c) -a*(a+b+c)*(a*a-b*c)
+(a*a-b*c)*(a*a-b*c) -(a*a+a*b+b*c)*(a*a+a*b+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 6 terms highest degree 2 of total 4
{ id3_6_2_4b = +a*a*(a+b+c)*(a+b+c) -a*a*(b-c)*(b-c)
+(a*b+a*c+b*c)*(a*b+a*c+b*c) -(a*a+a*c+b*c)*(a*a+a*c+b*c)
+(a*a-b*c)*(a*a-b*c) -(a*a+a*b+b*c)*(a*a+a*b+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Jahrbuch uber die Fortschritte der Mathematik, 1903, (p. 207)
\\ {} 3 vars with 6 terms highest degree 2 of total 4
{ id3_6_2_4c = +(a*a-b*b+c*c)*(a*a-b*b+c*c) -(a*a-b*b)*(a*a-b*b) -c*c*c*c
-c*c*(a+b)*(a+b) -c*c*(a-b)*(a-b) +(b*b+b*b)*(c*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Tito Piezas III, "https://math.stackexchange.com/q/2590555/".
\\ {} 3 vars with 6 terms highest degree 2 of total 4
{ id3_6_2_4d = +a*a*a*a +b*b*b*b -c*c*c*c -(a*a+b*b+c*c)*(a*a+b*b+c*c)
+(a*b-a*c+b*c+c*c)*(a*b-a*c+b*c+c*c)
+(a*b+a*c-b*c+c*c)*(a*b+a*c-b*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Edward J. Barbeau, Power Play, MAA, 1997. p.111 : "A Parametric solution
\\ to $ xyz = uvw $ and $ x^2+y^2+z^2=u^2+v^2+w^2 $ is given in (11) ..."
\\ {} 3 vars with 6 terms highest degree 2 of total 6
{ id3_6_2_6 = +b*b*(c*c-b*b+a*c)*(c*c-b*b+a*c)
+(a+c)*(a+c)*(b*b-c*c+a*c)*(b*b-c*c+a*c)
+a*a*b*b*(a-c)*(a-c) -a*a*b*b*(a+c)*(a+c)
-b*b*(b*b-c*c+a*c)*(b*b-c*c+a*c)
-(a-c)*(a-c)*(c*c-b*b+a*c)*(c*c-b*b+a*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 6 terms highest degree 4 of total 6
{ id3_6_4_6 = +a*a*(a*a-b*c)*(b*b+c*c) +(a*a-b*c)*(a*a-b*c)*(b*b+c*c)
-b*b*c*c*(a*a-b*b) -b*b*c*c*(a*a-c*c)
-(b+c)*(b+c)*(a*a-b*c)*(a*a-b*c)
-(b-c)*(b-c)*(a*a*a*a-a*a*b*c+b*b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Virginia M. Horak and Willis J. Horak, "Geometric Proofs of Algebraic
\\ Identities", The Mathematics Teacher, v. 74, n. 3 (1981) 212-216, p. 216.
\\ Katherine Stange, "Elliptic Nets and elliptic curves", arXiv:0710.13164v4
\\ (p. 22) : "f(x+y+z)-f(x+y)-f(y+z)-f(x+z)+f(x)+f(y)+f(z) = 0."
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (pp. 279,705)
\\ C. G. J. Jacobi, "Formulae Novae in Theoria Transcendentium Ellipticarum
\\ Fundamentales", in Werke, Bd. I, (1881) 333-341, p. 340. (before equ. (11.))
\\ {} 3 vars with 7 terms highest degree 1 of total 2
{ id3_7_1_2a = +(a+b+c)*(a+b+c) -(a+b)*(a+b) -(b+c)*(b+c)
-(a+c)*(a+c) +a*a +b*b +c*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Rober B. Nelson, "Proofs Without Words: Summing Squares by Counting
\\ Triangles", The College Mathematics Journal, v. 45, n. 8 (2015), p. 349.
\\ Replace {a -> -a+b+c, b -> a-b+c, c -> a+b-c} in id3_7_1_2a to get this.
\\ {} 3 vars with 7 terms highest degree 1 of total 2
{ id3_7_1_2b = +(a+a)*(a+a) +(b+b)*(b+b) +(c+c)*(c+c) -(a+b+c)*(a+b+c)
-(a+b-c)*(a+b-c) -(a-b+c)*(a-b+c) -(a-b-c)*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 7 terms highest degree 1 of total 2
{ id3_7_1_2c = +a*a +b*b +c*c +(a+a)*b +(a+a)*c +(b+b)*c -(a+b+c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 7 terms highest degree 4 of total 4
{ id3_7_4_4 = +a*b*(a-b)*(a-b) +a*c*(a-c)*(a-c) +b*c*(b-c)*(b-c)
+(a*a*b*b+a*a*c*c+b*b*c*c) +(a*b+a*c+b*c)*(a*b+a*c+b*c)
-(a*b+a*c+b*c)*(a*a+b*b+c*c) -a*b*c*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 8 terms highest degree 1 of total 3
{ id3_8_1_3 = +a*a*a +b*b*b +c*c*c +a*b*c +(a+b)*(a+c)*(b+c)
-a*(a+b)*(a+c) -b*(a+b)*(b+c) -c*(a+c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The n=3 case of an answer to "https://math.stackexchange.com/q/1346798/"
\\ {} 3 vars with 9 terms highest degree 1 of total 2
{ id3_9_1_2 = +a*a +b*b +c*c -a*b -a*c -b*c
+(a-b)*(c-a) +(a-b)*(b-c) +(c-a)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The IMO 2012 Problem 4 is based on this identity. If a+b+c=0 then
\\ f(a)^2+f(b)^2+f(c)^2 = 2f(a)f(b)+2f(b)f(c)+2f(c)f(a).
\\ {} 3 vars with 9 terms highest degree 2 of total 4 [CT]
{ id3_9_2_4 = +a*a*a*a +b*b*b*b +c*c*c*c -2*a*a*b*b -2*a*a*c*c -2*b*b*c*c
-(a+b+c)*(a+b+c)*(a+b+c)*(a+b+c) -8*a*b*c*(a+b+c)
+4*(a+b+c)*(a+b+c)*(a*b+a*c+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 3 vars with 10 terms highest degree 1 of total 3 [CT]
{ id3_10_1_3 = +a*a*a +b*b*b +c*c*c -3*a*b*c -a*(a+b)*(b-c) -a*(a-c)*(a+c)
+b*(a-b)*(a+b) +b*(a-c)*(b+c) -c*(a-b)*(a+c) +c*(b-c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 4 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is Ptolemy's theorem on distances between four points on a circle.
\\ This is the Cauchy Determinant case n=2 available at
\\ "http://dlmf.nist.gov/1.3#E14" and other sources.
\\ C. G. J. Jacobi, "Formulae Novae in Theoria Transcendentium Ellipticarum
\\ Fundamentales", in Werke, Bd. I, (1881) 333-341, p. 335. Section 1.
\\ Jim X. Xiang, "A Note on the Cauchy-Schwarz Inequality", A.M.M., (2013)
\\ v. 120, n. 5, (p. 456) : "$$ (c_1+c_2)(d_1+d_2) = (c_1+d_2)(d_1+c_2)+
\\ (c_2-d_2)(d_1-c_1). (4) $$"
\\ Wenchang Chu and Ying You, "Binomial Symmetries inspired by Bruckman's
\\ Problem", Filomat 24:1 (2010), 41-46, (p. 42 Theorem 1) case n=1:
\\ [[...]] Let x and y be two indeterminate and {\alpha_k,\gamma_k}_{k=1}^n
\\ complex numbers with {\gamma_k}_{k=1}^n being distinct. Then there hold
\\ the algebraic identity $$ \sum_{k=1}^n \frac{(\alpha_k-\gamma_k)(x-y)}
\\ {(x+\gamma_k)(y+\gamma_k)} \prod_{i=1, i\ne k}^n \frac{\alpha_i-\gamma_k}
\\ {\gamma_i-\gamma_k} = \prod_{i=1}^n\frac{y+\alpha_i}{y+\gamma_i}
\\ -\prod_{i=1}^n\frac{x+\alpha_i}{x+\gamma_i}. $$
\\ Max Alekseyev, Math Overflow question 280195, Sep 02 2017 case n=1:
\\ $$ (k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n
\\ \frac{t-d_j}{d_i-d_j}, $$
\\ {} 4 vars with 3 terms highest degree 1 of total 2 [TS]
{ id4_3_1_2a = +(a-b)*(c-d) -(a-c)*(b-d) +(a-d)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Shaun Cooper and Michael Hirschhorn, "On Some Infinite Product Identities",
\\ Rocky Mountain J. Math. 31 (2001), no. 1, 131-139, (p. 132 Lemma) case n=2:
\\ "[[...]] $$ [a;q]_\infty = (a;q)\infty(a^{{-1}q;q)\infty $$ [[...]] Lemma.
\\ Suppose $ a_1,a_2,\dots,a_n; b_1,b_2,\dots,b_n $ are nonzero complex numbers
\\ which satisfy (i) $ a_i\neq q^na_j $ for all $ i\neq j $ and all $ n\in Z $,
\\ (ii) $ a_1a_2\cdots a_n=b_1b_2\cdots b_n $. Then $$ \sum_{i=1}^n \frac{
\\ \prod_{j=1}^n[a_ib_j^{-1};q]_\infty}{\prod_{j=1,j\neq i}^n[a_ib_j^{-1};q]
\\ _\infty}}=0. $$"
\\ Replace [x y^{-1};q]_\infty with (x-y) throughout. Replace {a_1 -> c,
\\ a_2 -> b, b_1 -> a, b_2 -> d}, clear denominators and factor the sum of two
\\ terms giving (b-c)*(a-b-c+d).
\\ {} 4 vars with 3 terms highest degree 1 of total 2 [TS]
{ id4_3_1_2b = +(a-b)*(b-d) -(a-c)*(c-d) -(a-b-c+d)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Ivan Izmestiev, "Deformation of Quadrilaterals and Addition on Elliptic
\\ Curves", arXiv:1501.07157, p. 6, "$ ab-\bar{c}\bar{d}=(s-a-c)(s-b-c) $"
\\ where $ s=(a+b+c+d)/2, \bar{c}=s-c, \bar{d}=s-d $.
\\ {} 4 vars with 3 terms highest degree 1 of total 2 [TS]
{ id4_3_1_2c = +(a+a)*(b+b) -(a+b+c-d)*(a+b-c+d) +(a-b+c-d)*(a-b-c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The simplest four variable equation for the Weierstrass sigma function.
\\ This is the famous Weierstrass sigma function identity available at
\\ "http://dlmf.nist.gov/23.10#E4" and other sources.
\\ {} 4 vars with 3 terms highest degree 1 of total 4 [WS]
{ id4_3_1_4a = +(a+b)*(a-b)*(c+d)*(c-d)
+(b+c)*(b-c)*(a+d)*(a-d)
+(c+a)*(c-a)*(b+d)*(b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Katherine Stange, "Elliptic Nets and elliptic curves", p. 2, equ. (2).
\\ "W(p+q+s)W(p-q)W(r+s)W(r) + W(q+r+s)W(q-r)W(p+s)W(p) +
\\ W(r+p+s)W(r-p)W(q+s)W(q) = 0". Replace {p -> a, q -> b, r -> c, s -> -d}.
\\ Replace {a -> a+d, b -> b+d, c -> c+d, d -> d+d} to get id4_3_14a.
\\ {} 4 vars with 3 terms highest degree 1 of total 4 [WS]
{ id4_3_1_4b = +a*(b-c)*(a-d)*(b+c-d)
+b*(c-a)*(b-d)*(c+a-d)
+c*(a-b)*(c-d)*(a+b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Lucy Joan Slater, Generalized Hypergeometric Functions, 1996 (p. 204) :
\\ "[[...]] (7.4.4) can also be rewritten as $$ \vartheta_3(a)\vartheta_3(b+c)
\\ \vartheta_3(b+d)\vartheta_3(a+c+d) = \vartheta_3(b)\vartheta_3(a+c)
\\ \vartheta_3(a+d)\vartheta_3(b+c+d) + \vartheta_1(c)\vartheta_1(d)
\\ \vartheta_1(b-a)\vartheta_1(a+b+c+d). (7.4.5) $$".
\\ Replace {a -> b-c, b -> b-d, c -> c+d, d -> a-b} to get id4_3_1_4a.
\\ {} 4 vars with 3 terms highest degree 1 of total 4 [WS]
{ id4_3_1_4c = +a*(b+c)*(b+d)*(a+c+d)
-b*(a+c)*(a+d)*(b+c+d)
-c*d*(a-b)*(a+b+c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Buchstaber, Felder, Veselov, "Elliptic Dunkl operators, root systems,
\\ and functional equations". (p. 28, Proposition 19(iii)): "[[...]] $$
\\ \sigma_\lambda(z)\sigma_\mu(w) - \sigma_{\lambda+\mu}(w)\sigma_\lambda(z-w)
\\ - \sigma_\mu(w-z)\sigma_{\lambda+\mu}(z) = 0, $$".
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 715) :
\\ "[[...]], we see that (Knight) $$ (x-a)(x-b)(x-c)(x-a-b-c),
\\ x(x-a-b)(x-a-c)(x-b-c) $$ have three terms alike."
\\ Replace {a -> a-c+d, x -> d} to get this identity.
\\ Replace {a -> a-d, b -> b+d, c -> b-d, d -> b-c} to get id4_3_1_4a.
\\ {} 4 vars with 3 terms highest degree 1 of total 4 [WS]
{ id4_3_1_4d = +(a+b)*(a-c)*(b-d)*(c-d)
+a*d*(a+b-c)*(b+c-d)
-b*c*(a+b-d)*(a-c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 1 of total 4 [WS]
{ id4_3_1_4e = +b*(a-b)*(c-d)*(a-c-d)
-c*(a-c)*(b-d)*(a-b-d)
+d*(a-d)*(b-c)*(a-b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ R. W. H. T. Hudson, Kummer's quartic surface, Cambridge, 1905, p. 3.
\\ Alexy Gavrilov, "On the sigma function", arXiv:math/06031532v2, p. 1,
\\ equ. (2).
\\ Replace {a->(a+c)/2 , b->(a-c)/2, c->(b+d)/2, d->(b-d)/2} to get id4_3_1_4a.
\\ {} 4 vars with 3 terms highest degree 1 of total 4 [WS]
{ id4_3_1_4f = +(a-b-c+d)*(-a+b-c+d)*(-a-b+c+d)*(a+b+c+d)
+(-a+b+c+d)*(a-b+c+d)*(a+b-c+d)*(a+b+c-d)
-(a+a)*(b+b)*(c+c)*(d+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 1 of total 4
{ id4_3_1_4g = +(a-b)*(c-d)*(a+b-c-d)*(a+b-c-d)
-(a-c)*(b-d)*(a-b+c-d)*(a-b+c-d)
+(a-d)*(b-c)*(a-b-c+d)*(a-b-c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Zhi-Guo Liu, "A three-term theta function identity and its applications",
\\ Advances in Mathematics, 195 (2005), pp. 1-23. p. 4. "Theorem 4. We have
\\ $$\theta_1(z+x_1|\tau)\theta_1(z+x_2|\tau)\theta_1(z+x_3|\tau)
\\ \theta_1(z-x_1-x_2-x_3\tau)
\\ - \theta_1(z-x_1|\tau)\theta_1(z-x_2|\tau)\theta_1(z-x_3|\tau)
\\ \theta_1(z+x_1+x_2+x_3\tau)
\\ = -\theta_1(x_1+x_2|\tau)\theta_1(x_1+x_3|\tau)\theta_1(x_2+x_3|\tau)
\\ \theta_1(2z|\tau). (1.14) $$".
\\ {} 4 vars with 3 terms highest degree 1 of total 4 [WS]
{ id4_3_1_4h = +(a+b)*(a+c)*(a+d)*(a-b-c-d)
-(a-b)*(a-c)*(a-d)*(a+b+c+d)
+(a+a)*(b+c)*(b+d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into three rectangles.
\\ This is a part of Strassen multiplication of 2x2 matrices.
\\ This is used for Gauss multiplication of complex numbers. Reference at
\\ "http://en.wikipedia.org/wiki/Multiplication_algorithm"
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 7, equ. (3.4)) case n=1 : "Let u_k, v_k
\\ and w_k be three sequences, such that $$ u_k-v_k=w_k. $$ Then we have: $$
\\ \sum_k=0^n\frac{w_k}{w_0}\frac{u_0u_1\cdots u_{k-1}}{v_1v_2\cdots v_k}=
\\ \frac{u_0}{w_0}(\frac{u_1u2_\cdots u_n}{v_1v_2\cdots v_n}-\frac{v_0}{u_0}). $$"
\\ Christian Krattenthaler, "A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series", (p. 2): "Next take the identity
\\ (1-aq^k) - (1-a) = a(1-q^k)."
\\ Mark Haiman, Macdonald Polynomials and Geometry, 207-254 in New Perspectives
\\ in Algebraic Combinatorics, 1999. (p. 214, equ. (2.14)) case n=1 :
\\ "\prod_{i=1}^n \frac{1-t^{-1}zx_i}{1-zx_i} = t^{-n}+\sum_{i=1}^n \frac{1}
\\ {1-zx_i}\frac{\prod_{j=1}^n (1-x_j/tx_i)}{\prod_{j\ne i} (1-x_j/x_i)}"
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 16, equ. (1.3.6))
\\ case n=2 : "$$ c^n = \sum_{j=0}^n \frac{c^j(1/c)_j(q^{n+1-j})_j}{(q)_j} $$"
\\ Somos 2013: 1/x - (u-v)/(x*u-y*v) = -(1/x-1/y)*(y*v)/(x*u-y*v)
\\ {} 4 vars with 3 terms highest degree 2 of total 2 [NC]
{ id4_3_2_2a = +(a*b-c*d) -a*(b-d) -(a-c)*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 2 [NC]
{ id4_3_2_2b = +(a*b-c*d) -(a*d-c*b) -(a+c)*(b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011, (p. 31): "[[...]] or
\\ (1-b)(1-c)a -(1-a)(a-bc) = (a-b)(a-c)."
\\ {} 4 vars with 3 terms highest degree 2 of total 3
{ id4_3_2_3a = +a*(b-c)*(b-d) -b*(a-c)*(a-d) +(a-b)*(a*b-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 3
{ id4_3_2_3b = +a*(b-c)*(b+c) -b*(a*b-c*d) +c*(a*c-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 3
{ id4_3_2_3c = +a*b*(c-d) -(a*b-c*d)*c +(a*b-c*c)*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ If a(n) := q^n - 1, then a(n+m)*a(1) = a(n+1)*a(m) - q*a(m-1)*a(n) and is
\\ equivalent to (q^(n+m)-1)*(q-1) = (q^(n+1)-1)*(q^m-1) - (q^m-q)*(q^n-1).
\\ Replace {q^n -> x, q^m -> y, q -> z} to get (x*y-1)*(z-1) = (x*z-1)*(y-1)
\\ - (y-z)*(x-1). Replace {x -> b/a, y -> c/a, z -> d/a}.
\\ See Sagan and Savage, arXiv:0911.3159 page 4 Lemma 2.1 for the general
\\ identity "{n+m} = {n+1}{m} + t{m-1}{n}."
\\ {} 4 vars with 3 terms highest degree 2 of total 3
{ id4_3_2_3d = +a*(a-b)*(c-d) -(a-d)*(a*a-b*c) +(a-c)*(a*a-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 3
{ id4_3_2_3e = +a*b*(c-d) +d*(a*b+a*c+b*c+c*d) -c*(a+d)*(b+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 3
{ id4_3_2_3f = +a*b*b -(b+c)*(a*b+c*d) +c*(a*b+b*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 3
{ id4_3_2_3g = +(a+b)*(a+c)*(a+d) -(a+b+c+d)*(a*a+b*c+b*d+c*d)
+(b+c)*(b+d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4a = +a*b*(c*c+d*d) -c*d*(a*a+b*b) +(a*c-b*d)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4b = +a*(b-c)*(b*c-d*d) -b*(a-c)*(a*c-d*d) +c*(a-b)*(a*b-d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4c = +(a-b)*(c-d)*(a*b+c*d)
-(a-c)*(b-d)*(a*c+b*d)
+(a-d)*(b-c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4d = +(a-b)*(c+d)*(a*b+a*d+b*d)
-(a-c)*(b+d)*(a*c+a*d+c*d)
+(b-c)*(a+d)*(b*c+b*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From Wikipedia, "In algebra, the Brahmagupta-Fibonacci identity expresses
\\ the product of two sums of two squares as a sum of two squares in two
\\ different ways. ... The identity is also known as the Diophantus identity."
\\ From the norm of the product of two complex numbers.
\\ Lagrange's identity for n=2. Used to prove the Cauchy-Schwarz inequality.
\\ For example in https://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality.
\\ Also refer to https://en.wikipedia.org/wiki/Lagrange's_identity.
\\ This comes from Hadamard's Inequality case n=2 available at
\\ "http://dlmf.nist.gov/1.3#E8" and other sources.
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4e = +(a*c-b*d)*(a*c-b*d) +(a*d+b*c)*(a*d+b*c)
-(a*a+b*b)*(c*c+d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ B. C. Carlson, "Symmetry in c, d, n of Jacobian elliptic functions", J.
\\ Math. Anal. Appl. 299 (2004) 242-253, page 249 : "... The product of the
\\ numerators is $$ (ps_1qs_2rs_2-ps_2qs_1rs_1)(qs_1ps_2rs_2+qs_2ps_1rs_1) =
\\ ps_1qs_1ps_2qs_2(rs_2^2-rs_1^2)+(ps_1^2ps^2^2-ps^2^2qs_1^2)rs_1rs_2 ..."
\\ Let S_n := x^n-y^n. It satisfies identity $$ (S_n)^2 - S_{n+m}S_{n-m} =
\\ (xy)^(n-m)S_m^2. $$ Replace {x^n -> a, y^n -> b, x^m -> c, y^m -> d}.
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4f = +a*b*(c-d)*(c-d) -c*d*(a-b)*(a-b) +(a*c-b*d)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 16) case n=2 :
\\ "Aczel's Inequality. If $a,b$ are real n-tuples with $a_1^2-\sum_{i=2}^n
\\ a_i^2>0$ then $$ (a_1^2-\sum_{i=2}^n a_i^2)(b_1^2-\sum_{i=2}^n b_i^2)\le
\\ (a_1b_1-\sum_{i=2}^n a_ib_i)^2 $$, with equality if and only if $a~b$."
\\ Replace {a -> a*i, b -> b*i} where i^2 = -1 to get id4_3_2_4e.
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4g = +(a*c-b*d)*(a*c-b*d) -(a*d-b*c)*(a*d-b*c)
-(a-b)*(a+b)*(c-d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4h = +a*b*(a*d-b*c) +c*(b-d)*(a*b-a*d+c*d) -d*(a-c)*(a*b-b*c+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4i = +a*b*(a*b-a*d-b*c) +c*d*(a*d+b*c-c*d) -(a-c)*(b-d)*(a*b-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4j = +(c*c+d*d)*(a*b+c*d) +c*d*(a*a+b*b-c*c-d*d)
-(a*c+b*d)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4k = +(a*c+b*d)*(a*d+b*c) +a*b*(a*a+b*b-c*c-d*d)
-(a*a+b*b)*(a*b+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4l = +a*a*(b-d)*(b+d) -b*b*(a-c)*(a+c) +(a*d-b*c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This related to Ptolemy's inequality for a convex quadrilateral.
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4m = +(a+b+c-d)*(a+b-c+d)*(a-b+c+d)*(a-b-c-d)
-(a*a-b*b+c*c-d*d)*(a*a-b*b+c*c-d*d)
+(a*c+a*c+b*d+b*d)*(a*c+a*c+b*d+b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4n = +(a+b)*(a-c)*(b-d)*(c-d) -(a*b-a*c-b*c)*(b*c-b*d-c*d)
-(a*b-a*d-b*d)*(a*c-a*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 4
{ id4_3_2_4o = +(a-b)*(c-d)*(a*c+a*d-c*d) -(a-c)*(b-d)*(a*b+a*d-b*d)
+(a-d)*(b-c)*(a*b+a*c-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This leads to a solution to x^2 + a*b*y^2 = z^2 where x = a*v^2 -b*u^2,
\\ y = 2*u*v, z = a*v^2 +b*u^2. Replace {u^2 -> c, v^2 -> d}.
\\ {} 4 vars with 3 terms highest degree 2 of total 4 [CT]
{ id4_3_2_4p = +(a*d-b*c)*(a*d-b*c) +4*a*b*c*d -(a*d+b*c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 5
{ id4_3_2_5a = +(a-b)*(c-d)*(c-d)*(a*b-d*d)
-(a-c)*(b-d)*(b-d)*(a*c-d*d)
+(b-c)*(a-d)*(a-d)*(b*c-d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 5
{ id4_3_2_5b = +a*(b-c)*(b+c)*(a*d-b*c) +b*(a-c)*(a+c)*(a*c-b*d)
-c*(a-b)*(a+b)*(a*b-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 2 of total 6
{ id4_3_2_6 = +(a-b)*(a-c)*(b-c)*d*d*d
+(a*b+a*d+d*d)*(a*c+c*d+d*d)*(b*c+b*d+d*d)
-(a*b+b*d+d*d)*(a*c+a*d+d*d)*(b*c+c*d+d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 3 [NC]
{ id4_3_3_3a = -b*b*(c+d) -(a*a-b*b)*c +(a*a*c+b*b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 3
{ id4_3_3_3b = +(a+b)*(b*b+b*c-a*c) -b*b*(a+b+c+d) +(a*a*c+b*b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 3
{ id4_3_3_3c = +(a+b)*(a*a+a*d-b*d) -a*a*(a+b+c+d) +(a*a*c+b*b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 15, equ. (1.3.5))
\\ case n=2 : "$$ (a)_n = \sum_{k=0}^n (-1)^k\frac{(a{n+1-k})_k}{(q)_k}
\\ q^{k(k-1)/2}a^k. $$"
\\ {} 4 vars with 3 terms highest degree 3 of total 3
{ id4_3_3_3d = +b*a*(c+d) -(a+b)*(a*c+b*d) +(a*a*c+b*b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 3
{ id4_3_3_3e = +(a+b)*(c-d)*(c+d) -a*(c*c-c*d-d*d) -(a*c*d+b*c*c-b*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 4
{ id4_3_3_4a = +a*b*(b-c)*(b-d) -(a-b)*(b*b*b-a*c*d) -(b*b-a*c)*(b*b-a*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 4
{ id4_3_3_4b = +a*c*(b-d)*(b-d) -(a+c)*(a*d*d+b*b*c) +(a*d+b*c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ E. Whittaker and G. Robinson, "The Calculus of Observations", page 122,
\\ "Moreover, by Jacobi's theorem on the miors of the adjugate we have
\\ a_1 | a_1 a_2 a_2 ; a_0 a_1 a_2 ; 0; a_0 a_1 | = | a_1 a_2 ; a_0 a_1 ]^2
\\ - a_0^2 | a_2 a_3 ; a_1 a_2 |"
\\ {} 4 vars with 3 terms highest degree 3 of total 4
{ id4_3_3_4c = +a*a*(b*d-c*c) -b*(a*a*d-2*a*b*c+b*b*b) +(b*b-a*c)*(b*b-a*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 4
{ id4_3_3_4d = +(a-b)*(a*b*c+a*b*d+a*c*d+b*c*d)
-(a+c)*(a+d)*(b*c+b*d+c*d) +(b+c)*(b+d)*(a*c+a*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 4
{ id4_3_3_4e = +a*b*(c-d)*(c-d) +(a-b)*(a*d*d-b*c*c) -(a*d-b*c)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 5
{ id4_3_3_5 = +(a-b)*(c-d)*(a*a*b+a*b*b+c*c*d+c*d*d)
-(a-c)*(b-d)*(a*a*c+a*c*c+b*b*d+b*d*d)
+(a-d)*(b-c)*(a*a*d+a*d*d+b*b*c+b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 6
{ id4_3_3_6a = +a*b*c*d*(a-c)*(b-d)
+(a*b-c*d)*(a*b-c*d)*(a*b+c*d)
-(a*a*b-c*c*d)*(a*b*b-c*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 6
{ id4_3_3_6b = +a*a*b*b*(c-d)*(c-d)
+(a+b)*(a-b)*(a*c+b*d)*(a*c-b*d)
-(a*a*c-b*b*d)*(a*a*c-b*b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 7
{ id4_3_3_7a = +a*(b-c)*(b-c)*(b-d)*(a*a*b-c*c*d)
-b*(a-c)*(a-c)*(a-d)*(a*b*b-c*c*d)
+(a-b)*(a*b-c*c)*(a*b-c*d)*(a*b-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 3 of total 7
{ id4_3_3_7b = +a*(b-c)*(b+c)*(b-d)*(a*a*b+c*c*d)
-b*(a-c)*(a+c)*(a-d)*(a*b*b+c*c*d)
+(a-b)*(a*b+c*c)*(a*b-c*d)*(a*b+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 4 of total 6
{ id4_3_4_6c = +(a*a*a-b*c*d)*(a*a*a-b*c*d) +a*b*(a*a+a*c+b*c)*(a*a+a*d+c*d)
-(a*a+a*b+b*d)*(a*a*a*a+a*a*b*c+b*c*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Bruce Reznick, "Equal Sums of Two Cubes of Quadratic Forms" equ. (1.10)
\\ {} 4 vars with 3 terms highest degree 4 of total 10
{ id4_3_4_10 = +a*(b*c*c-a*c*d+b*d*d)*(b*c*c-a*c*d+b*d*d)*(b*c*c-a*c*d+b*d*d)
-b*(a*c*c-b*c*d+a*d*d)*(a*c*c-b*c*d+a*d*d)*(a*c*c-b*c*d+a*d*d)
+(a*a-b*b)*(a*c*c*c+b*d*d*d)*(b*c*c*c+a*d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 5 of total 9
{ id4_3_5_9 = +a*d*(b-d)*(c-d)*(a*a*a*b*c-d*d*d*d*d)
-(a+d)*(a*b-d*d)*(a*c-d*d)*(a*a*b*c-d*d*d*d)
+(a*a*b-d*d*d)*(a*a*c-d*d*d)*(a*b*c-d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 5 of total 10
{ id4_3_5_10a = +a*d*(a+d)*(b+d)*(c+d)*(a*a*a*b*c+d*d*d*d*d)
+(a-d)*(a*a*b-d*d*d)*(a*a*c-d*d*d)*(a*b*c-d*d*d)
-(a*a+d*d)*(a*b+d*d)*(a*c+d*d)*(a*a*b*c+d*d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 5 of total 10
{ id4_3_5_10b = +a*d*(b-d)*(b-d)*(c-d)*(a*a*b*b*c-d*d*d*d*d)
+(a-d)*(a*b*b-d*d*d)*(a*b*c-d*d*d)*(a*b*c-d*d*d)
-(a*b-d*d)*(a*b-d*d)*(a*c-d*d)*(a*b*b*c-d*d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 3 terms highest degree 5 of total 10
{ id4_3_5_10c = +a*d*(b-d)*(b+d)*(c-d)*(a*a*b*b*c+d*d*d*d*d)
+(a-d)*(a*b*b+d*d*d)*(a*b*c-d*d*d)*(a*b*c+d*d*d)
-(a*b-d*d)*(a*b+d*d)*(a*c-d*d)*(a*b*b*c+d*d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 562) :
\\ "Fauquembergue remarked that the last formula follows from the identity
\\ $$ (y_1^3-y^3)(x_1^2y-x^2y_1)^3 + (x^3-x_1^3)(y_1^2x-y^2x_1)^3 =
\\ (x^3y_1^3-y^3x_1^3)(xy-x_1y_1)^3, $$ due to A. Desboves, [[...]]"
\\ {} 4 vars with 3 terms highest degree 6 of total 12
{ id4_3_6_12 = +(a*a*a-b*b*b)*(a*d*d-b*c*c)*(a*d*d-b*c*c)*(a*d*d-b*c*c)
+(a*a*d-b*b*c)*(a*a*d-b*b*c)*(a*a*d-b*b*c)*(c*c*c-d*d*d)
-(a*a*a*d*d*d-b*b*b*c*c*c)*(a*c-b*d)*(a*c-b*d)*(a*c-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id4_3_2_2 with its first term split into two terms.
\\ This is the difference of two overlapping rectangles equals the difference
\\ of the two rectangles with the common rectangular intersection removed.
\\ B. C. Berndt, Ramanujan's Notebooks, Part IV, (p. 37, (26.1)) case r=2 :
\\ "$$ \sum_{j=0}^{r-1} (a_j-b_{j+1})\left(\frac{b_1\dots b_j}{a_1\dots a_j}
\\ \right) = a_0 -\left(\frac{b_1\dots b_r}{a_1\dots a_{r-1}}\right). $$
\\ J. M. Steele, The Cauchy-Schwarz Master Class, 2004 (p. 276) case n=2 :
\\ "Solution for Exercise 12.8. [[...]] Here the telescoping identity $$
\\ a_1a_2\cdots a_n-b_1b_2\cdots b_n = \sum_{j=1}^n a_1\cdots a_{j-1}(a_j-b_j)
\\ b_{j+1}\cdots b_n $$ makes the Weierstrass inequality immediate."
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC]
{ id4_4_1_2a = +a*b -c*d -a*(b-d) -(a-c)*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ B. C. Berndt, Ramanujan's Notebooks, Part IV, (p. 36., (25.1)) case n=2 :
\\ "$$ \sum_{k=0}^{n-1} \frac{a_0a_1\dots a_k}{(x+a_1)(x+a_2)\dots(x+a_{k+1})}
\\ = \frac{1}{x}-\frac{1}{x(1+x/a_1)(1+x/a_2)\dots(1+x/a_n)}. $$"
\\ Replace {a -> b-a, b -> a-d, c -> a-c, d -> c-a} in id4_4_1_2m.
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC]
{ id4_4_1_2b = +(a-b)*(a-c) +(b-d)*(a-b) -(a-d)*(a-c) +(b-d)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC]
{ id4_4_1_2c = +(a-b)*(a-b) +(a-b)*(b-c) +(b-d)*(a-c) -(a-d)*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC]
{ id4_4_1_2d = +(a-b)*(a-c) +(a-b)*(c-d) -(a-c)*(a-d) +(b-c)*(a-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC]
{ id4_4_1_2e = +(a-c)*b -a*(b-c) -c*(a-d) +c*(b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC]
{ id4_4_1_2f = +(a+b+c)*d -(a+b-c)*d +c*(a-b-d) -c*(a-b+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. E. Andrews and D. Zeilberger, "A Short Proof of Jacobi's Formula for
\\ the number of representations of an integer as a sum of four squares",
\\ arXiv:math/9206203v1. Equation (1) 2nd part is equivalent to:
\\ q^k(1-q^{n+1}) - (1+q^{n+1}) = - (1+q^k)q^{n+1} - (1-q^k).
\\ Replace {q^k -> a/b, q^{n+1} -> c/d} to get this identity
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC]
{ id4_4_1_2g = +a*(c-d) +b*(c+d) -(a+b)*c +(a-b)*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [TS]
{ id4_4_1_2h = +a*(b-c+d) -b*(a-c+d) -c*(b-a+d) +d*(b-a+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Arthur Cayley, A Trigonometrical Identity, Messenger of Math., 7 (1878),
\\ p. 124. "cos(b-c) cos(b+c+d) + cos a cos(a+d) = cos(c-a) cos(c+a+d) +
\\ cos b cos(b+d) = cos(a-b) cos(a+b+d) + cos c cos(c+d) = cos a cos(a+d) +
\\ cos b cos(b+d) + cos c cos(c+d) - cos d."
\\ Replace {a -> a-pi, b -> b-pi, d -> -d} in the first equation to convert
\\ the cos a to sin a, for example.
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [TS]
{ id4_4_1_2i = +a*(a-d) -(a-c)*(a+c-d) -b*(b-d) +(b-c)*(b+c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [TS]
{ id4_4_1_2j = +(a-b)*(c-d) +(a-d)*(b-c) +(a+b)*(c+d) -(a+d)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 'slaw', https://math.stackexchange.com/q/3762472, Jul 19, 2020
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [NC,CT]
{ id4_4_1_2k = +(a-c)*(b+d) +(a+c)*(b-d) -2*a*b +2*c*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Ivan Izmestiev, "Deformation of Quadrilaterals and Addition on Elliptic
\\ Curves", arXiv:1501.07157, p. 6, "$ ab+cd=\bar{a}\bar{b}+\bar{c}\bar{d} $"
\\ where $ s=(a+b+c+d)/2, \bar{a}=s-c, \bar{b}=s-d, ... $.
\\ {} 4 vars with 4 terms highest degree 1 of total 2 [TS]
{ id4_4_1_2l = +(a+a)*(b+b) -(a+b+c-d)*(a+b-c+d) +(a-b-c-d)*(a-b+c+d)
+(c+c)*(d+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Ivan Izmestiev, "Deformation of Quadrilaterals and Addition on Elliptic
\\ Curves", arXiv:1501.07157, p. 12, "... using the fact that $a+b=c+d$
\\ implies $$ ac-bd=(a+b)(a-d), cd-ab=(a-c)(a-d) $$".
\\ {} 4 vars with 4 terms highest degree 1 of total 2
{ id4_4_1_2m = +a*c -b*d -(a+b)*(a-d) +a*(a+b-c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957, Chapter XXII, Art.
\\ 310, p. 254, "_Example_. Prove that $$ \frac{(x-b)(x-c)}{(a-b)(a-c)} +
\\ \frac{(x-c)(x-a)}{(b-c)(b-a)} + \frac{(x-a)(x-b)}{(c-a)(c-b)} = 1. $$"
\\ E. W. Hobson, A Treatise on Plane and Advanced Trigonometry, 7th edition,
\\ 1928, Dover Publication reprint 1957, Chapter XV, Art. 255, p. 312, the
\\ Example assumes the Hall and Knight identity.
\\ Doron Zeilberger, "Ptolemy-type Theorem on four points on a circle" at
\\ "http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/sadov.html"
\\ T. Amdeberhan, O. Espinosa, V. Moll, A. Straub, "Wallis-Ramanujan-Schur-
\\ Feynman", (p.5, equ. (4.3)) case n=3 : "\prod_{k=1}^n \frac{1}{y+b_k} =
\\ \sum_{k=1}^n \frac{1}{y+b_k}\prod_{j=1,j\ne k}^n\frac{1}{b_j-b_k}. "
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=4, r=0 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ J. W. L. Glaisher, "On certain formulae in elliptic functions", vol. XIX,
\\ 1883, Quarterly Journal of Pure and Applied Mathematics, pp. 22-37. page 33:
\\ " may be written $$ sn(w-y)sn(w-z)sn(y-z) +sn(w-z)sn(w-x)sn(z-x)
\\ +sn(w-x)sn(w-y)sn(x-y) +sn(y-z)sn(z-x)sn(x-y) = 0; $$ "
\\ {} 4 vars with 4 terms highest degree 1 of total 3 [JE]
{ id4_4_1_3a = +(a-b)*(a-c)*(b-c) -(a-b)*(a-d)*(b-d)
+(a-c)*(a-d)*(c-d) -(b-c)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3
{ id4_4_1_3b = +(a-b)*(a-c)*(b-c) -a*(b-c)*(a-d)
+b*(a-c)*(b-d) -c*(a-b)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3 [TS]
{ id4_4_1_3c = +a*a*(c-d) -b*b*(c-d) +(a-b)*(a-c)*(b+d) -(a-b)*(a-d)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3 [TS]
{ id4_4_1_3d = +a*(b-c)*(b+c) -a*(b-d)*(b+d) +c*(a-d)*(c-d) +d*(a+c)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3 [TS]
{ id4_4_1_3e = +a*a*(c-d) -b*c*(b-d) +b*d*(b-c) -(a-b)*(a+b)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gasper+Rahman, Basic Hypergeometric Series 2nd ed. (p. 331, equ. (11.7.3))
\\ case n=2 : "[[...]] is the easily verifiable identity
\\ \sum_{k=1}^n\frac{\prod_{j=1}^n(b_j-a_k)}{a_k\prod_{j\ne k}(a_j-a_k)} =
\\ \frac{b_1...b_n}{a_1...a_n}-1. "
\\ {} 4 vars with 4 terms highest degree 1 of total 3
{ id4_4_1_3f = +a*b*(a-b) -c*d*(a-b) +a*(b-c)*(b-d) -b*(a-c)*(a-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3 [TS]
{ id4_4_1_3g = +(a-b)*(a-c)*(a+c) -(a-b)*(a-d)*(a+d)
-(b-d)*(c-d)*(a+c) +(a-d)*(c-d)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3
{ id4_4_1_3h = +(a-b)*(a-b)*(c-d) -(a-c)*(a-c)*(b-d)
+(a-d)*(a-d)*(b-c) -(b-c)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is Stewart's theorem for four points on a line available at
\\ "http://en.wikipedia.org/wiki/Stewart's_theorem" and other sources.
\\ "$$ PA^2\cdot BC+PB^2\cdot CA+PC^2\cdot AB+BC\cdot CA\cdot AB=0 $$"
\\ {} 4 vars with 4 terms highest degree 1 of total 3
{ id4_4_1_3i = +(a-d)*(a-d)*(b-c) +(b-d)*(b-d)*(c-a)
+(c-d)*(c-d)*(a-b) +(b-c)*(c-a)*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3 [TS]
{ id4_4_1_3j = +a*b*(c-d) -a*c*(b-d) +b*d*(a-c) -c*d*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 3 [TS]
{ id4_4_1_3k = +(a-b)*b*d +(a+b-c)*(b-d)*c
+(a-c)*(b-c)*(b-d) -(a-d)*b*b ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=4, r=1 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ Equivalent to $$ x = \frac{(x-a)(x-b)}{(c-a)(c-b)}c +
\\ \frac{(x-b)(x-c)}{(a-b)(a-c)}a + \frac{(x-a)(x-c)}{(b-a)(b-c)}b $$
\\ {} 4 vars with 4 terms highest degree 1 of total 4
{ id4_4_1_4a = +a*(b-c)*(b-d)*(c-d) -b*(a-c)*(a-d)*(c-d)
+c*(a-b)*(a-d)*(b-d) -d*(a-b)*(b-c)*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 4 [TS]
{ id4_4_1_4b = +(a-b)*(a+b)*(c-d)*b -(a-c)*(a+c)*(b-d)*c
+(a-d)*(a+d)*(b-c)*d +(b-c)*(b-d)*(c-d)*(b+c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 4 [TS]
{ id4_4_1_4c = +a*a*(b-c)*(b-d) -b*b*(a-c)*(a-d)
+a*d*(a-b)*(b-c) +b*c*(a-b)*(a-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 257, Examples 12(1)).
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=4, r=2 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 4 vars with 4 terms highest degree 1 of total 5 [TS]
{ id4_4_1_5a = +a*a*(b-c)*(b-d)*(c-d) -b*b*(a-c)*(a-d)*(c-d)
+c*c*(a-b)*(a-d)*(b-d) -d*d*(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 5 [TS]
{ id4_4_1_5b = +(a-d)*(a-d)*b*c*(b-c) -(b-d)*(b-d)*a*c*(a-c)
+(c-d)*(c-d)*a*b*(a-b) -d*d*(a-b)*(a-c)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 1 of total 6 [WS]
{ id4_4_1_6a = +a*b*(a-b)*(c-d)*(a-c-d)*(b-c-d)
-a*c*(a-c)*(b-d)*(a-b-d)*(b-c-d)
-b*c*(a-d)*(b-c)*(a-b+d)*(a-c-d)
+(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J. W. L. Glaisher, "On certain formulae in elliptic functions", vol. XIX,
\\ 1883, Quarterly Journal of Pure and Applied Mathematics, pp. 22-37. page 33.
\\ {} 4 vars with 4 terms highest degree 1 of total 6 [JE]
{ id4_4_1_6b = +(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d)
-(a-b)*(a+c)*(a+d)*(b+c)*(b+d)*(c-d)
+(a+b)*(a-c)*(a+d)*(b+c)*(b-d)*(c+d)
-(a+b)*(a+c)*(a-d)*(b-c)*(b+d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is part of Karatsuba multiplication of double precision numbers,
\\ and with sign changes, multiplication of complex numbers (Gauss?).
\\ {} 4 vars with 4 terms highest degree 2 of total 2 [NC]
{ id4_4_2_2a = +a*c +b*d -(a-b)*(c-d) -(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. E. Andrews, R. P. Lewis, "An algebraic identity of F. H. Jackson and its
\\ implications for partitions", Disc. Math. 232 (2001) 77-83. (p. 79, equ.
\\ (1.9)) : "Setting $d=0$ in (1.8) to get the simple $$ [[...]] = 1 +\frac
\\ {b}{1-b} +\frac{c}{1-c} = \frac{1-bc}{(1-b)(1-c)} $$".
\\ {} 4 vars with 4 terms highest degree 2 of total 2 [NC]
{ id4_4_2_2b = +(a*b-c*d) -c*(b-d) -(a-c)*(b-d) -(a-c)*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 2 of total 2 [NC]
{ id4_4_2_2c = +(a*b-c*d) -c*(b-d) -(a-c)*(b-c) -(a-c)*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 2 of total 3 [NC]
{ id4_4_2_3a = +a*b*(c+d) +(a-b)*c*d +b*c*d -a*(b*c+b*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 2 of total 3
{ id4_4_2_3b = +a*a*(c-d) -a*(a-b)*(c-d) +b*d*(a-b) -b*(a*c-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 2 of total 3
{ id4_4_2_3c = +a*b*c +d*(a*b+a*c+b*c+c*d) -a*b*d -c*(a+d)*(b+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ case n=2 of $$ \frac{1}{\prod_{i=1}^n 1-xq^{i-1}} = 1 +
\\ \sum_{k=1}^n \frac{xq^{k-1}}{\prod_{i=1}^k 1-xq^{i-1}} $$.
\\ {} 4 vars with 4 terms highest degree 2 of total 3
{ id4_4_2_3d = -a*a*d +a*b*c +b*(a*d-b*c) +(a-b)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From $ 1/x+1/y+1/z-1/(x+y+z) = (x+y)(x+z)(y+z)/(xyz(x+y+z)) $.
\\ {} 4 vars with 4 terms highest degree 2 of total 3
{ id4_4_2_3e = +a*b*(c+d) +(a+b)*c*d +(b+c)*(b+d)*(c+d)
-(a+b+c+d)*(b*c+b*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From $ x^{k+1}+1/x^{k+1} = (x^k+1/x^k)(x+1/x) - (x^{k-1}+1/x^{k-1}) $.
\\ Replace {t -> a, u -> -b, p -> d, q -> c} in case n=2 :
\\ $$ \prod_{k=0}^{n-1} (u + (q/p)^k t) = \sum_{k=0}^n t^k u^{n-k}
\\ (q/p)^{k(k-1)/2} {n \choose k}_{q/p}. $$
\\ {} 4 vars with 4 terms highest degree 2 of total 3
{ id4_4_2_3f = +a*a*c +b*b*d -a*b*(c+d) -(a-b)*(a*c-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 712) :
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4a = +(a*b+b*d+c*d)*(a*b+b*d+c*d) +(a*b+a*c+c*d)*(a*b+a*c+c*d)
-(a*b+b*d+c*d)*(a*b+a*c+c*d) -(a*a+a*d+d*d)*(b*b+b*c+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 58) : "follows
\\ from the identity $$ \frac{|z|^2}{p} +\frac{|w|^2}{q} -\frac{|z+w|^2}{p+q}
\\ = \frac{|q z-p q|^2}{pq(p+q)}. $$"
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4b = +a*d*d*(a+b) +b*c*c*(a+b)
-a*b*(c+d)*(c+d) -(a*d-b*c)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Harold M. Edwards, A Normal Form for Elliptic Curves, B.A.M.S, v. 44 n. 3,
\\ p. 403, equation (8.3) : "(1-P^2)^2 = [[[...]]] = (1+P^2)(1+x^2y^2)(1+x_1^2)
\\ (1+y_1^2)-x^2y^2(1+x_1^2y_1^2)^2-x_1^2y_1^2(1+x^2y^2)^2"
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4c = +a*c*(b+d)*(b+d) +b*d*(a+c)*(a+c)
-(a*d+c*b)*(a+c)*(b+d) +(a*d-b*c)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Albert H. Beiler, Recreations in the Theory of Numbers, Dover Publications,
\\ 1964, p. 143, (Formula 3).
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4d = +(a*c+b*d)*(a*c+b*d) +(a*d-b*c)*(a*d-b*c)
-(a*c-b*d)*(a*c-b*d) -(a*d+b*c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4e = +(a-b)*(c-d)*(a*d+b*c) +(a-b)*(c-d)*(a*c+b*d)
+(a+b)*(c+d)*(a*d+b*c) -(a+b)*(c+d)*(a*c+b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From x/(1+x) + y/(1+y) < (x+y)/(1+x+y) if x and y > 0.
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4f = +a*b*(a*c+b*d+c*d+c*d) -a*(b+c)*(a*c+b*d+c*d)
+(a+d)*(b+c)*(a*c+b*d) -b*(a+d)*(a*c+b*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From the norm of product of two (a+b*w) where 0=1+w+w^2.
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4g = +(a*c-b*d)*(a*c-b*d) -(a*c-b*d)*(a*d+b*c-b*d)
+(a*d+b*c-b*d)*(a*d+b*c-b*d) -(a*a+b*b-a*b)*(c*c+d*d-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From det(A*A^T) = det(A^T*A) for a 2x2 matrix A.
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4h = +(a*a-b*b)*(c*c-d*d) -(a*a-c*c)*(b*b-d*d)
+(a*b-c*d)*(a*b-c*d) -(a*c-b*d)*(a*c-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4i = +a*c*(b-d)*(b-d) +b*d*(a+c)*(a+c)
-(a*d+b*c)*(a+c)*(b+d) +(a*d+b*c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 222) :
\\ "Consider the integral squares equal to the sums of the squares of three
\\ integers; for instance $(m^2+n^2+p^2+q^2)^2 = (m^2+n^2-p^2-q^2)^2 +
\\ (2mp+2nq)^2 + (2mq-2np)^2."
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4j =
+(a*a+b*b+c*c+d*d)*(a*a+b*b+c*c+d*d)
-(a*a+b*b-c*c-d*d)*(a*a+b*b-c*c-d*d)
-(a*c+a*c+b*d+b*d)*(a*c+a*c+b*d+b*d)
-(a*d+a*d-b*c-b*c)*(a*d+a*d-b*c-b*c)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J.-P. Allouche, G. Pirillo and L. Vannucci, "How a simple questions on the
\\ Fibonacci sequence lead the authors to a number-theoretical promenade",
\\ http://www-igm.univ-mlv.fr/~berstel/Colloque-Pitagore/all-pir-van-march2020.pdf
\\ p. 8, Lemma 6.2 :
\\ "Proof. It suffices to use any of the three following identities
\\ (y^2+xy-x^2)(b^2+ab-a^2) = (xb+yb-xa)^2 + (xb+yb-xa)(ay-bx) - (ay-bx)^2"
\\ {} 4 vars with 4 terms highest degree 2 of total 4
{ id4_4_2_4k = +(a*d-b*c)*(a*d-b*c) +(a*d-b*c)*(a*c-b*c-b*d)
-(a*c-b*c-b*d)*(a*c-b*c-b*d) +(a*a-a*b-b*b)*(c*c-c*d-d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 2 of total 5
{ id4_4_2_5 = +a*a*(b-c)*(b*c+b*d+c*d) +b*(a*a-c*d)*(c*c-b*d)
-b*c*d*(b-c)*(b+c+d) -c*(a*a-b*d)*(b*b-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. E. Andrews and D. Zeilberger, "A Short Proof of Jacobi's Formula for
\\ the number of representations of an integer as a sum of four squares",
\\ arXiv:math/9206203v1. Equation (1) 1st part is equivalent to:
\\ q^k(1-q^{n+1})(1+q^{n+1})^3(1+q^{n+k+1})(q^k+q^{n+1})
\\ -q^k(1+q^{n+1})(1-q^{n+1})^3(1-q^{n+k+1})(q^k-q^{n+1}) =
\\ q^{n+1}(1+q^k)^2(1+q^{2n+2})(q^k-q^{n+1})(1+q^{n+k+1})
\\ -q^{n+1}(1+q^k)^2(1+q^{2n+2})(q^k+q^{n+1})(1-q^{n+k+1}).
\\ Replace {q^{n+1} -> b/d, q^k -> -c/a} to get this identity.
\\ {} 4 vars with 4 terms highest degree 2 of total 10
{ id4_4_2_10 =
+a*c*(b-d)*(b+d)*(b+d)*(b+d)*(a*b-c*d)*(a*d-b*c)
+a*c*(b-d)*(b-d)*(b-d)*(b+d)*(a*b+c*d)*(a*d+b*c)
+b*d*(a-c)*(a-c)*(b*b+d*d)*(a*b+c*d)*(a*d-b*c)
-b*d*(a-c)*(a-c)*(b*b+d*d)*(a*b-c*d)*(a*d+b*c)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 3 of total 4
{ id4_4_3_4 = +a*d*(b-d)*(c-d) -(a-d)*(c+d)*(a*b-d*d)
+(a-c)*d*(a*b-d*d) +(a-d)*(a*b*c-d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 16, equ. (1.3.6))
\\ case n=3 : "$$ c^n = \sum_{j=0}^n \frac{c^j(1/c)_j(q^{n+1-j})_j}{(q)_j} $$"
\\ {} 4 vars with 4 terms highest degree 3 of total 5
{ id4_4_3_5a = -d*d*d*(a*a+a*b+b*b) +b*b*d*(c*c+c*d+d*d)
+b*(a*d-b*c)*(c*c+c*d+d*d) +(a*d-b*c)*(a*d*d-b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 3 of total 5
{ id4_4_3_5b = +(a+b)*c*d*(c-d)*(c+d) -a*c*d*(c*c-c*d-d*d)
+(a*c*d+b*c*c-b*d*d)*(c*c-c*d-d*d)
-(a*c*d+b*c*c-b*d*d)*(c-d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, New First Course in the Theory of Equations, Wiley,
\\ 1939. (p. 119, Prob. 6) [4x4 variant Vandermonde determinant].
\\ {} 4 vars with 4 terms highest degree 3 of total 6
{ id4_4_3_6a = +b*b*c*c*(a-d)*(b-c) -b*b*d*d*(a-c)*(b-d) +c*c*d*d*(a-b)*(c-d)
-(b-c)*(b-d)*(c-d)*(a*b*c+a*b*d+a*c*d-b*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Shaun Cooper, The Quintuple product identity, Int. J. Number Theory 2(2006),
\\ no. 1, 115-161. (equ. (43)) case n=1 : "Let $$f(n,k)=(-1)^kq^{k(3k+1)/2}
\\ x^{3k} [2n n+k]_q \frac{1-x^2q^{2k+1}}{(q^2x^2;q)_{n+k}(x^{-2};q)_{n-k}},
\\ [[...]] Then $$ \sum_{k=-n}^n f(n,k)=\frac{1-x^2q}{(xq;q)_n(x^{-1};q)_n}. $$"
\\ {} 4 vars with 4 terms highest degree 3 of total 6
{ id4_4_3_6b = +a*a*(a*b-c*d)*(a*b+c*d) +a*c*(b+d)*(a*a*b-c*c*d)
+c*c*d*d*(a-c)*(a+c) -(a+c)*(a*b+c*d)*(a*a*b-c*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 4 terms highest degree 3 of total 6
{ id4_4_3_6c = +a*b*c*d*(a-c)*(b-d) +a*(b-d)*(a*b-c*d)*(a*b+c*d)
+d*(a-c)*(a*b-c*d)*(a*b+c*d) -(a*a*b-c*c*d)*(a*b*b-c*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ W. Y. C. Chen, W. Chu, N. S. S. Gu, Finite form of the quintuple product
\\ identity, J. of Comb. Theory, Ser. A, 113 (2006), pp. 185-187, (p. 286)
\\ case m=2 : "Theorem. (Finite form of the quintuple product identity). For
\\ a variable x, there holds an algebraic identity: $$ 1 = \sum_{k=0}^m
\\ (1+xq^k)[m k]\frac{(x;q)_{m+1}}{(q^kx^2;q)_{m+1}}x^kq^k^2. $$"
\\ {} 4 vars with 4 terms highest degree 4 of total 8
{ id4_4_4_8 = +a*(b+d)*(a*b-c*d)*(a*d*d*d-b*b*b*c)
+b*b*b*b*c*(a-c)*(a*b-c*d)
-b*(a-c)*(b+d)*(a*d+b*c)*(a*d*d-b*b*c)
+c*(a*d*d-b*b*c)*(a*d*d*d-b*b*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 15, equ. (1.3.5))
\\ case n=3 : "$$ (a)_n = \sum_{k=0}^n (-1)^k\frac{(a{n+1-k})_k}{(q)_k}
\\ q^{k(k-1)/2}a^k. $$"
\\ {} 4 vars with 4 terms highest degree 6 of total 6
{ id4_4_6_6 = +a*a*b*c*(c*c+c*d+d*d) +a*b*b*d*(c*c+c*d+d*d)
-(a+b)*(a*c+b*d)*(a*c*c+b*d*d) +(a*a*a*c*c*c+b*b*b*d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id4_4_2_2b with its first term split into two terms.
\\ George E. Andrews, The Theory of Partitions, 1984, (p. 209, Example 3) :
\\ "The proof relies on the crucial observation that $$ \frac{1}{1-xu} +\frac
\\ {1}{1-yu} -1 =\frac{1-xyu^2}{(1-xu)(1-yu)} $$ is a special case of"
\\ {} 4 vars with 5 terms highest degree 1 of total 2 [NC]
{ id4_5_1_2a = +a*b -c*d -c*(b-d) -(a-c)*(b-d) -(a-c)*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id4_4_2_2c with its first term split into two terms.
\\ {} 4 vars with 5 terms highest degree 1 of total 2
{ id4_5_1_2b = +a*b -c*d -c*(b-d) -(a-c)*(b-c) -(a-c)*c ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 1 of total 3
{ id4_5_1_3 = +(a-d)*(b-d)*(c-d) -(a-b)*(a-c)*(a-d) +(a-b)*(b-c)*(b-d)
-(a-c)*(b-c)*(c-d) +(a+b-c-d)*(a-b+c-d)*(a-b-c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 1 of total 4
{ id4_5_1_4a = +(a-b)*(a-d)*(a-d)*(b-d) +(b-c)*(b-d)*(b-d)*(c-d)
-(a-c)*(a-d)*(c-d)*(c-d) +(a-d)*(b-c)*(b-c)*(a-b-c+d)
-(a-b)*(a-c)*(b-d)*(a+b-c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 1 of total 4 [TS]
{ id4_5_1_4b = +(a-b)*(a-b)*(a+b)*(a+b) +(c-d)*(c-d)*(c+d)*(c+d)
-(a-b)*(a-b)*(c+d)*(c+d) -(a+b)*(a+b)*(c-d)*(c-d)
-(a+b-c-d)*(a-b+c-d)*(a-b-c+d)*(a+b+c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. W. Turnbull, The theory of deteminants, matrices, and invariants, 3rd
\\ ed., Dover, 1960. (p. 28 EXAMPLES): " 1. Prove (d-c)(d-b)(d-a)(c-b)(c-a)
\\ (b-a) = | 1 1 1 1 \\ a b c d \\ a^2 b^2 c^2 d^2 \\ a^3 b^3 c^3 d^3 |.
\\ This determinant is called an alternant."
\\ {} 4 vars with 5 terms highest degree 1 of total 6
{ id4_5_1_6a = +a*b*c*(a-b)*(a-c)*(b-c) -a*b*d*(a-b)*(a-d)*(b-d)
+a*c*d*(a-c)*(a-d)*(c-d) -b*c*d*(b-c)*(b-d)*(c-d)
-(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id4_4_3_6 with its last term split into two terms.
\\ {} 4 vars with 5 terms highest degree 1 of total 6 [TS]
{ id4_5_1_6b = +b*b*c*c*(a-d)*(b-c)
-b*b*d*d*(a-c)*(b-d)
+c*c*d*d*(a-b)*(c-d)
-(b-c)*(b-d)*(c-d)*a*b*(c+d)
-(b-c)*(b-d)*(c-d)*c*d*(a-b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 1 of total 6
{ id4_5_1_6c = +a*a*a*(b-c)*(b-d)*(c-d) -b*b*b*(a-c)*(a-d)*(c-d)
+c*c*c*(a-b)*(a-d)*(b-d) -d*d*d*(a-b)*(a-c)*(b-c)
-(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gasper+Rahman, Basic Hypergeometric Series 2nd ed. (p. 152, Ex. 5.27):
\\ "Prove that S(a^{-2},bc,bd,cd)/S(b/a,c/a,d/a,abcd) + idem(a;b,c,d) = 2,
\\ where S is defined by Ex. 2.16."
\\ {} 4 vars with 5 terms highest degree 1 of total 7
{ id4_5_1_7a = +(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d)*(a+b+c+d)
-a*(b-c)*(b+c)*(b-d)*(b+d)*(c-d)*(c+d)
+b*(a-c)*(a+c)*(a-d)*(a+d)*(c-d)*(c+d)
-c*(a-b)*(a+b)*(a-d)*(a+d)*(b-d)*(b+d)
+d*(a-b)*(a+b)*(a-c)*(a+c)*(b-c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id4_5_1_7a multiplied by 2 but with 2*a -> a+a and so on.
\\ {} 4 vars with 5 terms highest degree 1 of total 7 [WS,CT]
{ id4_5_1_7b = +2*(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d)*(a+b+c+d)
-(a+a)*(b-c)*(b+c)*(b-d)*(b+d)*(c-d)*(c+d)
+(b+b)*(a-c)*(a+c)*(a-d)*(a+d)*(c-d)*(c+d)
-(c+c)*(a-b)*(a+b)*(a-d)*(a+d)*(b-d)*(b+d)
+(d+d)*(a-b)*(a+b)*(a-c)*(a+c)*(b-c)*(b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 1 of total 7
{ id4_5_1_7c = +a*b*(c-d)*(a+b-c-d)*(a-b-c-d)*(a+b+c+d)*(a-b+c+d)
+a*d*(b-c)*(a+b+c-d)*(a-b-c-d)*(a+b+c+d)*(a-b-c+d)
+a*d*(b+c)*(a-b+c-d)*(a+b-c-d)*(a-b+c+d)*(a+b-c+d)
-b*c*(a+d)*(a+b+c-d)*(a-b+c-d)*(a+b-c-d)*(a-b-c-d)
-b*d*(a-c)*(a+b+c-d)*(a-b+c-d)*(a+b+c+d)*(a-b+c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 1 of total 7 [TS]
{ id4_5_1_7d = +a*a*a*a*(b-c)*(b-d)*(c-d)
-b*b*b*b*(a-c)*(a-d)*(c-d)
+c*c*c*c*(a-b)*(a-d)*(b-d)
-d*d*d*d*(a-b)*(a-c)*(b-c)
-(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d)*(a+b+c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J. H. Silverman, The Arithmetic of Dynamical Systems, Springer, 2007.
\\ p. 190, "We also note the formal identity $$ (X-1)^2-(XY-1)(XZ-1) =
\\ X(X+Y+Z-2-XYZ) $$
\\ {} 4 vars with 5 terms highest degree 2 of total 4
{ id4_5_2_4a = +a*a*(a-b)*(a-b) -(a*a-b*c)*(a*a-b*d) +a*a*b*(a-b)
-b*c*(a*a-b*d) +a*a*b*(a-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. M. Stark, An Introduction to Number Theory, Markham, 1970, p. 163,
\\ "10. Show that if $ x^5 + y^5 + z^5 = 0, $ then $ 2(x + y + z)^5 =
\\ 5(x + y)(x + z)(y + z)[(x + y + z)^2 + x^2 + y^2 + z^2]. $" Replace
\\ {x -> a-b, y -> b-c, z -> c-d}.
\\ {} 4 vars with 5 terms highest degree 2 of total 5 [CT]
{ id4_5_2_5 = +(a-b)*(a-b)*(a-b)*(a-b)*(a-b) -(a-d)*(a-d)*(a-d)*(a-d)*(a-d)
+(b-c)*(b-c)*(b-c)*(b-c)*(b-c) +(c-d)*(c-d)*(c-d)*(c-d)*(c-d)
+5*(a-c)*(b-d)*(a-b+c-d)*(a*a+b*b+c*c+d*d-a*b-a*d-b*c-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Question at "https://math.stackexchange.com/q/2801902/" to solve
\\ $$ \frac{x+a^2}{a+b}+\frac{x+b^2}{b+c}+\frac{x+c^2}{c+a}=2(a+b+c) $$ with
\\ solution $ x = ab+bc+ca $. Let $ d = -a-b-c. $
\\ {} 4 vars with 5 terms highest degree 3 of total 4
{ id4_5_3_4a = +(a+b)*(a+c)*(b+c)*(d+d)
+(a+b)*(b+c)*(a*b-c*d)
+(a+b)*(a+c)*(a*c-b*d)
-(a+c)*(b+c)*(a*d-b*c)
-(a+b+c+d)*(a*a*c+a*b*b+b*c*c+a*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Question at "https://math.stackexchange.com/q/4874899/": "Show that if
\\ $x + y + z = xy + xz + yz$ then $x ^ 2 * (1 - y ^ 2) + y ^ 2 * (1 - z ^ 2)
\\ + z ^ 2 * ( 1 - x ^ 2) = 2(x + y + z)(xyz - 1)$" answer by 'gpmath'.
\\ {} 4 vars with 5 terms highest degree 3 of total 4 [CT]
{ id4_5_3_4b = +a*a*(d*d-b*b) +b*b*(d*d-c*c) +c*c*(d*d-a*a)
+(a*b+a*c+b*c-a*d-b*d-c*d)*(a*b+a*c+b*c+a*d+b*d+c*d+d*d+d*d)
-2*(a+b+c)*(a*b*c-d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 3 of total 5 [CT]
{ id4_5_3_5 = +(a+a)*(b*b-d*d)*(c*c-d*d)
+(b+b)*(a*a-d*d)*(c*c-d*d)
+(c+c)*(a*a-d*d)*(b*b-d*d)
-(a+a)*(b+b)*(c+c)*d*d
-2*(a*b+a*c+b*c-d*d)*(a*b*c-a*d*d-b*d*d-c*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. E. Andrews, R. P. Lewis, "An algebraic identity of F. H. Jackson and its
\\ implications for partitions", Disc. Math. 232 (2001) 77-83. (p. 79, equ.
\\ (1.10)) : "... and, setting $ d=bc $ in (1.8), we find $$ [[...]] = 1 +
\\ \frac{b}{1-b} +\frac{c}{1-c} +\frac{bc}{1-b^2c^2} = \frac{(1-b^2c)(1-bc^2)}
\\ {(1-b)(1-c)(1-b^2c^2)}. $$"
\\ {} 4 vars with 5 terms highest degree 3 of total 6
{ id4_5_3_6a = +a*b*c*d*(a-c)*(b-d)
+c*(b-d)*(a*b-c*d)*(a*b+c*d)
+d*(a-c)*(a*b-c*d)*(a*b+c*d)
+(a-c)*(b-d)*(a*b-c*d)*(a*b+c*d)
-(a*a*b-c*c*d)*(a*b*b-c*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 3 of total 6
{ id4_5_3_6b = +a*b*(a-c)*(a-d)*(a*a-c*d)
+a*c*(a-b)*(a-d)*(a*a-b*d)
+a*d*(a-b)*(a-c)*(a*a-b*c)
+(a-b)*(a-c)*(a-d)*(a*a*a+b*c*d)
-(a*a-b*c)*(a*a-b*d)*(a*a-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ case n=3 of $$ \frac{1}{\prod_{i=1}^n 1-xq^{i-1}} = 1 +
\\ \sum_{k=1}^n \frac{xq^{k-1}}{\prod_{i=1}^k 1-xq^{i-1}} $$.
\\ {} 4 vars with 5 terms highest degree 3 of total 6
{ id4_5_3_6c = -a*a*a*d*d*d +a*a*b*c*c*d +a*b*c*(a*d*d-b*c*c)
+b*(a*d-b*c)*(a*d*d-b*c*c)
+(a-b)*(a*d-b*c)*(a*d*d-b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {t -> a, u -> -b, p -> c, q -> d} in case n=3 :
\\ $$ \prod_{k=0}^{n-1} (u + (q/p)^k t) = \sum_{k=0}^n t^k u^{n-k}
\\ (q/p)^{k(k-1)/2} {n \choose k}_{q/p}. $$
\\ {} 4 vars with 5 terms highest degree 3 of total 6
{ id4_5_3_6d = +a*a*a*d*d*d -a*a*b*d*(c*c+c*d+d*d)
+a*b*b*c*(c*c+c*d+d*d) -b*b*b*c*c*c
-(a-b)*(a*d-b*c)*(a*d*d-b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Routh's theorem. See Wikipedia article. x -> b/a, y -> c/a, z-> d/a in
\\ "$$ 1 - \frac{x}{zx+x+1} - \frac{y}{xy+y+1} - \frac{z}{yz+z+1} =
\\ \frac{(xyz-1)^2}{(xz+x+1)(yx+y+1)(zy+z+1)}. $$"
\\ {} 4 vars with 5 terms highest degree 3 of total 6
{ id4_5_3_6e = +a*b*(a*a+a*c+b*c)*(a*a+a*d+c*d)
+a*c*(a*a+a*b+b*d)*(a*a+a*d+c*d)
+a*d*(a*a+a*b+b*d)*(a*a+a*c+b*c)
+(a*a*a-b*c*d)*(a*a*a-b*c*d)
-(a*a+a*b+b*d)*(a*a+a*c+b*c)*(a*a+a*d+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Alexander Berkovich, "The Tri-Pentagonal Number Theorem and Related
\\ Identities", (pub. in Intl. J. of Num. Th., 5 (2009) 1385-1399) Jan 2008.
\\ arXiv: math.NT/0801.3008 (p. 4, equ. (2.1)): "[[...]]
\\ (1-x_1)(1-x_1q) +(1-x_2)(1-x_2q) -(1-x_1x_2q)(1-x_1x_2q^2) -(1-x_1)(1-x_1q)
\\ (1-x_2)(1-x_2q) +x_1x_2(1+q)(1-x_1q)(1-x_2q) = 0".
\\ {} 4 vars with 5 terms highest degree 4 of total 7
{ id4_5_4_7a = +a*a*a*a*(a-b)*(a*a-b*d)
+a*a*a*a*(a-c)*(a*a-c*d)
-a*(a-b)*(a-c)*(a*a-b*d)*(a*a-c*d)
+b*c*(a+d)*(a*a-b*d)*(a*a-c*d)
-(a*a*a-b*c*d)*(a*a*a*a-b*c*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 5 terms highest degree 4 of total 7
{ id4_5_4_7b = +c*c*d*d*(a-b)*(a*a-b*d)
+b*b*d*d*(a-c)*(a*a-c*d)
-d*(a-b)*(a-c)*(a*a-b*d)*(a*a-c*d)
+a*a*(a+d)*(a*a-b*d)*(a*a-c*d)
-(a*a*a-b*c*d)*(a*a*a*a-b*c*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 16, equ. (1.3.6))
\\ case n=4 : "$$ c^n = \sum_{j=0}^n \frac{c^j(1/c)_j(q^{n+1-j})_j}{(q)_j} $$"
\\ {} 4 vars with 5 terms highest degree 4 of total 9
{ id4_5_4_9 = -d*d*d*d*d*d*(a*a*a+a*a*b+a*b*b+b*b*b)
+b*b*b*d*d*d*(c*c*c+c*c*d+c*d*d+d*d*d)
+b*b*d*(a*d-b*c)*(c*c+d*d)*(c*c+c*d+d*d)
+b*(c*c*c+c*c*d+c*d*d+d*d*d)*(a*d-b*c)*(a*d*d-b*c*c)
+(a*d-b*c)*(a*d*d-b*c*c)*(a*d*d*d-b*c*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Kevin S. Brown, "https://www.mathpages.com/home/kmath071.htm" :"there's a
\\ parametric solution: (6 t^3 + 1)^3 - (6 t^3 - 1)^3 - (6 t^2)^3 = 2".
\\ {} 4 vars with 5 terms highest degree 4 of total 12 [CT]
{ id4_5_4_12 = +(a*a*a*d+b*b*b*c)*(a*a*a*d+b*b*b*c)*(a*a*a*d+b*b*b*c)
+(a*a*a*d-b*b*b*c)*(a*a*a*d-b*b*b*c)*(a*a*a*d-b*b*b*c)
-a*a*a*b*b*b*b*b*b*c*c*c
-2*a*a*a*a*a*a*a*a*a*d*d*d
+a*a*a*b*b*b*b*b*b*c*c*(c-d-d-d-d-d-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 15, equ. (1.3.5))
\\ case n=4 : "$$ (a)_n = \sum_{k=0}^n (-1)^k\frac{(a{n+1-k})_k}{(q)_k}
\\ q^{k(k-1)/2}a^k. $$"
\\ {} 4 vars with 5 terms highest degree 10 of total 10
{ id4_5_10_10 = +a*a*a*b*c*c*c*(c+d)*(c*c+d*d)
+a*a*b*b*c*d*(c*c+d*d)*(c*c+c*d+d*d)
+a*b*b*b*d*d*d*(c+d)*(c*c+d*d)
-(a+b)*(a*c+b*d)*(a*c*c+b*d*d)*(a*c*c*c+b*d*d*d)
+(a*a*a*a*c*c*c*c*c*c+b*b*b*b*d*d*d*d*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the determinant of the 3x3 rank 2 matrix
\\ [a-b, c+d, c-d; b+c, a-d, a+d; a-c, b+d, b-d]
\\ {} 4 vars with 6 terms highest degree 1 of total 3 [JE]
{ id4_6_1_3 = +(a-b)*(a-d)*(b-d) -(a-b)*(a+d)*(b+d) +(a-c)*(a+d)*(c+d)
-(a-c)*(a-d)*(c-d) +(b+c)*(b+d)*(c-d) -(b+c)*(b-d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 6 terms highest degree 1 of total 5 [JE]
{ id4_6_1_5a = +a*c*d*(a-c)*(a-d) -b*c*d*(a-d)*(b-c) -a*b*b*(a-b)*(a-b)
-b*b*d*(a-b)*(b-d) +b*(a-b)*(a-b)*(a-d)*(b-d)
+d*(a-b)*(a-c)*(a-d)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 6 terms highest degree 1 of total 5 [JE]
{ id4_6_1_5b = +a*b*d*(a-b)*(a-d) -a*b*c*(a-b)*(a-c) -b*c*d*(a-c)*(b-d)
+b*c*d*(a-d)*(b-c) +c*(a-b)*(a-c)*(a-d)*(b-d)
-d*(a-b)*(a-c)*(a-d)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 6 terms highest degree 1 of total 5 [JE]
{ id4_6_1_5c = +(a-b)*(a-b)*(a-d)*(b-d)*(b-d) -(a-c)*(a-c)*(a-d)*(c-d)*(c-d)
+(a-b)*(b-c)*(b-d)*(b-d)*(c-d) +(a-c)*(b-c)*(b-d)*(c-d)*(c-d)
-(a-b)*(a-b)*(a-c)*(b-c)*(b-d) -(a-b)*(a-c)*(a-c)*(b-c)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Derived from identity s(a+b)s(a-b) = (s(a)^2-s(b)^2)/(1-(ks(a)s(b))^2).
\\ {} 4 vars with 6 terms highest degree 1 of total 8 [JE,ZS]
{ id4_6_1_8 = +a*a*b*b*c*c*(a-b)*(a+b) -a*a*b*b*d*d*(a-b)*(a+b)
-a*a*b*b*(a-b)*(a+b)*(c-d)*(c+d) -a*a*c*c*d*d*(c-d)*(c+d)
+b*b*c*c*d*d*(c-d)*(c+d) +c*c*d*d*(a-b)*(a+b)*(c-d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 266) :
\\ {} 4 vars with 6 terms highest degree 2 of total 4
{ id4_6_2_4b = +a*a*b*b +c*c*d*d -a*a*d*d -b*b*c*c +(a*d+b*c)*(a*d+b*c)
-(a*b+c*d)*(a*b+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ MR1798572 Andrews, George E.; Lewis, Richard; Liu, Zhi-Guo
\\ "An identity relating a theta function to a sum of Lambert series".
\\ Bull. London Math. Soc. 33 (2001), no. 1, 25-31. (p. 27, equ. (9)) :
\\ "We shall use the elementary identity (see [2]) $$ 1 +\frac{a}{1-a} +\frac
\\ {b}{1-b} +\frac{c}{1-c} -\frac{abc}{1-abc} = \frac{(1-bc)(1-ac)(1-ab)}
\\ {(1-a)(1-b)(1-c)(1-abc)}."
\\ G. E. Andrews, R. P. Lewis, "An algebraic identity of F. H. Jackson and its
\\ implications for partitions", Disc. Math. 232 (2001) 77-83. (p. 78, equ.
\\ (1.8)): "... and then set $a=1$, we find $$ 1 +\frac{b}{1-b} +\frac{c}{1-c}
\\ +\frac{d}{1-d} -\frac{bcd}{1-bcd} = \frac{(1-bc)(1-bd)(1-cd)}
\\ {(1-b)(1-c)(1-d)(1-bcd)}. $$"
\\ {} 4 vars with 6 terms highest degree 3 of total 6
{ id4_6_3_6a = +a*b*c*(a-d)*(b-d)*(c-d)
-a*(b-d)*(c-d)*(a*b*c-d*d*d)
-b*(a-d)*(c-d)*(a*b*c-d*d*d)
-c*(a-d)*(b-d)*(a*b*c-d*d*d)
+(a-d)*(b-d)*(c-d)*(a*b*c-d*d*d)
+(a*c-d*d)*(a*b-d*d)*(b*c-d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 6 terms highest degree 3 of total 6
{ id4_6_3_6b = +d*d*d*(a-d)*(b-d)*(c-d)
-d*(b-d)*(c-d)*(a*b*c-d*d*d)
-d*(a-d)*(c-d)*(a*b*c-d*d*d)
-d*(a-d)*(b-d)*(a*b*c-d*d*d)
-(a-d)*(b-d)*(c-d)*(a*b*c-d*d*d)
+(a*c-d*d)*(a*b-d*d)*(b*c-d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Ramanujan's identity: f(a+b+c)+f(b+c+d)+f(a-d) = f(a+b+d)+f(a+c+d)+f(b-c)
\\ when ad=bc and f(x)=x^2 or x^4. Replace {a->a/b, b->d/c, c->a/d, d->b/c}
\\ {} 4 vars with 6 terms highest degree 3 of total 6
{ id4_6_3_6c = +(a*b*c+a*c*d+b*d*d)*(a*b*c+a*c*d+b*d*d)
+b*b*(a*c+b*d+d*d)*(a*c+b*d+d*d)
+d*d*(a*c-b*b)*(a*c-b*b)
-d*d*(a*c+b*b+b*d)*(a*c+b*b+b*d)
-(a*b*c+a*c*d+b*b*d)*(a*b*c+a*c*d+b*b*d)
-b*b*(a*c-d*d)*(a*c-d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 16, equ. (1.3.6))
\\ case n=5 : "$$ c^n = \sum_{j=0}^n \frac{c^j(1/c)_j(q^{n+1-j})_j}{(q)_j} $$"
\\ {} 4 vars with 6 terms highest degree 5 of total 14
{ id4_6_5_14 =
-d*d*d*d*d*d*d*d*d*d*(a*a*a*a+a*a*a*b+a*a*b*b+a*b*b*b+b*b*b*b)
+b*b*b*b*d*d*d*d*d*d*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
+b*b*b*d*d*d*(c*c+d*d)*(a*d-b*c)*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
+b*b*d*(a*d-b*c)*(a*d*d-b*c*c)*(c*c+d*d)*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
+b*(a*d-b*c)*(a*d*d-b*c*c)*(a*d*d*d-b*c*c*c)*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
+(a*d-b*c)*(a*d*d-b*c*c)*(a*d*d*d-b*c*c*c)*(a*d*d*d*d-b*c*c*c*c)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 6 terms highest degree 8 of total 10
{ id4_6_8_10 = +a*b*b*b*c*c*(c*c*c*c+d*d*d*d)
-a*b*b*c*c*c*d*(a+b)*(c*c+d*d)
-b*c*c*d*d*(a*a+b*b)*(a*d*d-b*c*c)
-d*d*(a*a*a*a*d*d*d*d+b*b*b*b*c*c*c*c)
+a*c*d*d*d*(a+b)*(a*a*d*d+b*b*c*c)
-a*(c-d)*(a*d-b*c)*(a*d*d-b*c*c)*(a*d*d+b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Andrews and B. Berndt, Lost Notebook Part III, (p. 15, equ. (1.3.5))
\\ case n=5 : "$$ (a)_n = \sum_{k=0}^n (-1)^k\frac{(a{n+1-k})_k}{(q)_k}
\\ q^{k(k-1)/2}a^k. $$"
\\ {} 4 vars with 6 terms highest degree 15 of total 15
{ id4_6_15_15 =
+a*a*a*a*b*c*c*c*c*c*c*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
+a*a*a*b*b*c*c*c*d*(c*c+d*d)*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
+a*a*b*b*b*c*d*d*d*(c*c+d*d)*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
+a*b*b*b*b*d*d*d*d*d*d*(c*c*c*c+c*c*c*d+c*c*d*d+c*d*d*d+d*d*d*d)
-(a+b)*(a*c+b*d)*(a*c*c+b*d*d)*(a*c*c*c+b*d*d*d)*(a*c*c*c*c+b*d*d*d*d)
+(a*a*a*a*a*c*c*c*c*c*c*c*c*c*c+b*b*b*b*b*d*d*d*d*d*d*d*d*d*d)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 202) :
\\ "Parallelogram Inequality. If $a,b,c,d$ are real $n$-tuples, then
\\ $$ |a-b|^2+|b-c|^2+|c-d|^2+|d-a|^2\ge|a-c|^2+|d-b|^2, (1) $$ with
\\ equality if and only if [[..]] are coplanar and form a parallelogram."
\\ {} 4 vars with 7 terms highest degree 1 of total 2 [NC]
{ id4_7_1_2 = +(a-b)*(a-b) +(b-c)*(b-c) +(c-d)*(c-d) +(a-d)*(a-d)
-(a-c)*(a-c) -(b-d)*(b-d) -(a-b+c-d)*(a-b+c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 7 terms highest degree 1 of total 4
{ id4_7_1_4 = +(a-b)*(a-d)*(a-d)*(b-d) +(b-c)*(b-d)*(b-d)*(c-d)
-(a-c)*(a-d)*(c-d)*(c-d) +(a-d)*(b-c)*(b-c)*(a-b)
-(a-d)*(b-c)*(b-c)*(c-d) -(a-b)*(a-c)*(b-d)*(a-c)
-(a-b)*(a-c)*(b-d)*(b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Derived from Brahmagupta's formula for the area of a cyclic quadrilateral.
\\ {} 4 vars with 7 terms highest degree 2 of total 4
{ id4_7_2_4 = +(a+a)*a*a*a +(b+b)*b*b*b +(c+c)*c*c*c +(d+d)*d*d*d
-(a*a+b*b+c*c+d*d)*(a*a+b*b+c*c+d*d) -(a+a)*(b+b)*(c+c)*d
-(a-b+c+d)*(a+b+c-d)*(a+b-c+d)*(a-b-c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 421, Examples 24).
\\ {} 4 vars with 7 terms highest degree 4 of total 6
{ id4_7_4_6 = +(a*b+c*d)*(b*b*c*c+a*a*d*d)
-(a*b+c*d)*(c*c*a*a+b*b*d*d)
+(a*c+b*d)*(a*a*b*b+c*c*d*d)
-(a*c+b*d)*(b*b*c*c+a*a*d*d)
+(a*d+b*c)*(c*c*a*a+b*b*d*d)
-(a*d+b*c)*(a*a*b*b+c*c*d*d)
+(b-c)*(c-a)*(a-b)*(a-d)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Alice Simon and Peter Volkmann, "On Two Geometric Inequalities", Annales
\\ Mathematicae Silesianae 9 (1995), pp. 137-140. (p. 139, equ. (7)).
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 278,705) :
\\ "Goldbach (p. 526) noted that $ \alpha, \beta,\ gamma, \alpha+\beta+\gamma
\\ +2\delta, and \alpha+\gamma+delta, \beta+\gamma+\delta, \delta and
\\ \alpha+\delta, \beta+\delta, \gamma+\delta, \alpha+\beta+\gamma+\delta $
\\ have the same sum of squares."
\\ {} 4 vars with 8 terms highest degree 1 of total 2
{ id4_8_1_2a = +(a+b+d)*(a+b+d) +(a+c+d)*(a+c+d) +(b+c+d)*(b+c+d) +d*d
-(a+d)*(a+d) -(b+d)*(b+d) -(c+d)*(c+d) -(a+b+c+d)*(a+b+c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 8 terms highest degree 1 of total 2
{ id4_8_1_2b = +(a+b+c+d)*(a+b+c+d) +(a+b-c-d)*(a+b-c-d) +(a-b+c-d)*(a-b+c-d)
+(a-b-c+d)*(a-b-c+d) -(a-b-c-d)*(a-b-c-d) -(a-b+c+d)*(a-b+c+d)
-(a+b-c+d)*(a+b-c+d) -(a+b+c-d)*(a+b+c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 8 terms highest degree 1 of total 2
{ id4_8_1_2c = +(a+a)*(a+a) +(b+b)*(b+b) +(c+c)*(c+c) +(d+d)*(d+d)
-(a-b-c-d)*(a-b-c-d) -(a-b+c+d)*(a-b+c+d)
-(a+b-c+d)*(a+b-c+d) -(a+b+c-d)*(a+b+c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 8 terms highest degree 2 of total 4
{ id4_8_2_4 = +a*a*d*d -a*a*b*(a-d) -a*a*(a-d)*(c-d) +a*a*(a+b-c)*(b+c-d)
-(a*b-c*d)*(a*b-c*d) -a*d*(a+b-c)*(a+b+c) +a*d*(a+b-c)*(a+b-c)
+c*c*(a-d)*(a-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 9 terms highest degree 1 of total 2
{ id4_9_1_2 = +(a-b)*(a-b) +(a-c)*(a-c) +(a-d)*(a-d) -(a-b)*(a-c)
-(a-c)*(a-d) -(a-b)*(a-d) -b*(b-c) -c*(c-d) +d*(b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 4 vars with 10 terms highest degree 1 of total 2
{ id4_10_1_2 = +(a-b)*(a-b) -(a-b)*(a-d) +(a-b)*(b-c) -(a-c)*(a-c)
+(a-d)*(a-d) -(a-d)*(c-d) +(b-c)*(b-c) +(b-c)*(c-d)
-(b-d)*(b-d) +(c-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 5 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 1 of total 2 [TS]
{ id5_3_1_2 = +(a-b)*(c-d) -(a-c-e)*(b-d-e) +(a-d-e)*(b-c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Shaun Cooper and Michael Hirschhorn, "On Some Infinite Product Identities",
\\ Rocky Mountain J. Math. 31 (2001), no. 1, 131-139, (p. 132 Lemma) case n=3:
\\ "[[...]] $$ [a;q]_\infty = (a;q)\infty(a^{{-1}q;q)\infty $$ [[...]] Lemma.
\\ Suppose $ a_1,a_2,\dots,a_n; b_1,b_2,\dots,b_n $ are nonzero complex numbers
\\ which satisfy (i) $a_i\neq q^na_j$ for all $i\neq j$ and all $n\in Z$, (ii)
\\ $a_1a_2\cdots a_n=b_1b_2\cdots b_n$. Then $$ \sum_{i=1}^n \frac{ \prod_{j=1}
\\ ^n[a_ib_j^{-1};q]_\infty}{\prod_{j=1,j\neq i}^n[a_ib_j^{-1};q]_\infty}}=0. $$"
\\ Replace [x y^{-1};q]_\infty with (x-y) throughout. Replace {a_1 -> a,
\\ a_2 -> b, a_3 -> c, b_1 -> d, b_2 -> e, b_3 -> a+b+c-d-e}.
\\ Replace {e -> 0, d -> -d} to get id4_3_1_4b.
\\ {} 5 vars with 3 terms highest degree 1 of total 4 [WS]
{ id5_3_1_4 = +(a-b)*(c-d)*(c-e)*(a+b-d-e)
-(a-c)*(b-d)*(b-e)*(a+c-d-e)
+(a-d)*(a-e)*(b-c)*(b+c-d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 2 of total 3
{ id5_3_2_3a = +a*(b-c)*(d-e) -(a-c)*(a*d-b*e) +(a-b)*(a*d-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 2 of total 3
{ id5_3_2_3b = +a*(b*d-c*c) -b*(a*d-c*e) +c*(a*c-b*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 2 of total 4
{ id5_3_2_4a = +a*a*(b-c)*(d-e) -(a*a-c*d)*(a*a-b*e) +(a*a-b*d)*(a*a-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 2 of total 4
{ id5_3_2_4b = +(a*c-b*b)*(c*e-d*d) -(a*d-b*c)*(b*e-c*d) +(a*e-b*d)*(b*d-c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Shaun Cooper and Michael Hirschhorn, "On Some Infinite Product Identities",
\\ Rocky Mountain J. Math. 31 (2001), no. 1, 131-139, (p. 132 Lemma) case n=3:
\\ "[[...]] $$ [a;q]_\infty = (a;q)\infty(a^{{-1}q;q)\infty $$ [[...]] Lemma.
\\ Suppose $ a_1,a_2,\dots,a_n; b_1,b_2,\dots,b_n $ are nonzero complex numbers
\\ which satisfy (i) $a_i\neq q^na_j$ for all $i\neq j$ and all $n\in Z$, (ii)
\\ $a_1a_2\cdots a_n=b_1b_2\cdots b_n$. Then $$ \sum_{i=1}^n \frac{ \prod_{j=1}
\\ ^n[a_ib_j^{-1};q]_\infty}{\prod_{j=1,j\neq i}^n[a_ib_j^{-1};q]_\infty}}=0. $$"
\\ Replace [x;q]_\infty with (1-x) throughout. Replace {a_1 -> a, a_2 -> b,
\\ a_3 -> c, b_1 -> d, b_2 -> e, b_3 -> abc/(de)}.
\\ {} 5 vars with 3 terms highest degree 2 of total 5
{ id5_3_2_5 = +(a-b)*(c-d)*(c-e)*(a*b-d*e) -(a-c)*(b-d)*(b-e)*(a*c-d*e)
+(a-d)*(a-e)*(b-c)*(b*c-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gasper+Rahman, Basic Hypergeometric Series 2nd ed. (p. 303):
\\ "[[...]] which is simply the elliptic analogue of the trivial identity
\\ $$ (1-x\lambda)(1-x/\lambda)(1-\mu\nu)(1-\mu/\nu)
\\ -(1-x\nu)(1-x/\nu)(1-\lambda\mu)(1-\mu/\lambda) = \frac{\mu,\lambda}
\\ (1-x\mu)(1-x/\mu)(1-\lambda\nu)(1-\lambda/\nu). (11.1.1) $$"
\\ {} 5 vars with 3 terms highest degree 2 of total 6
{ id5_3_2_6a = +(a-b)*(c-d)*(a*b-e*e)*(c*d-e*e)
-(a-c)*(b-d)*(a*c-e*e)*(b*d-e*e)
+(a-d)*(b-c)*(a*d-e*e)*(b*c-e*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 2 of total 6
{ id5_3_2_6b = +(a-b)*(a+b)*(b+b)*(c-d)*(c-e)*(d-e)
-(a*c-b*d+b*e)*(b*c+a*d-b*e)*(b*c-b*d-a*e)
+(a*c+b*d-b*e)*(b*c-a*d-b*e)*(b*c-b*d+a*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 3 of total 4
{ id5_3_3_4 = +a*e*(b-d)*(c-d) -(a*b-d*e)*(a*c-d*e) +(a-e)*(a*b*c-d*d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From (a+bI)(a+cI)(d+eI) = (aad-abe-ace-bcd)+(aae+abd+acd-bce)I where I^2=-1.
\\ {} 5 vars with 3 terms highest degree 3 of total 6
{ id5_3_3_6 = +(a*a+b*b)*(a*a+c*c)*(d*d+e*e)
-(a*a*d-a*b*e-a*c*e-b*c*d)*(a*a*d-a*b*e-a*c*e-b*c*d)
-(a*a*e+a*b*d+a*c*d-b*c*e)*(a*a*e+a*b*d+a*c*d-b*c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p.32, equ. (9.8)) :
\\ (1-b)(1-c)(1-d)(a^2-bcd)a -(1-a)(a-bc)(a-bd)(a-cd) = (a-b)(a-c)(a-d)(a-bcd)
\\ {} 5 vars with 3 terms highest degree 3 of total 7
{ id5_3_3_7 = +a*(b-c)*(b-d)*(b-e)*(a*a*b-c*d*e)
-b*(a-c)*(a-d)*(a-e)*(a*b*b-c*d*e)
+(a-b)*(a*b-c*d)*(a*b-c*e)*(a*b-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 4 of total 5
{ id5_3_4_5a = +a*(a*b*b*c-a*d*d*e-b*c*e*e)
-b*(a*a*b*c+b*b*d*e-c*d*d*e)
+e*(a*a*d*d+a*b*c*e+b*b*b*d-b*c*d*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 4 of total 5
{ id5_3_4_5b = +a*(a*b*c*e+a*a*d*d+b*b*b*d-b*c*d*d)
-e*(a*a*b*c+b*b*d*e-c*d*d*e)
-d*(a*a*a*d+a*b*b*b-a*b*c*d-b*b*e*e+c*d*e*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 3 terms highest degree 4 of total 5
{ id5_3_4_5c = +b*(a*c-e*e)*(c*d-e*e) -c*(a*b-e*e)*(b*d-e*e)
+(b-c)*(a*b*c*d-e*e*e*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ E. Whittaker and G. Robinson, "The Calculus of Observations", page 122,
\\ "Again, by Jacobi's theorem we have
\\ |a_1 a_2 ; a_0 a_1| |a_1 a_2 a_3 a_4 ; a_0 a_1 a_2 a_3 ;
\\ 0 a_0 a_1 a_2 ; 0 0 a_0 a_1| = |a_1 a_2 a_3 ; a_0 a_1 a_2 ; 0 a_0 a_1|^2
\\ -a_0^3 |a_2 a_3 a_4 ; a_1 a_2 a_3 ; a_0 a_1 a_2 |"
\\ {} 5 vars with 3 terms highest degree 4 of total 6
{ id5_3_4_6 = +a*a*a*(a*c*e-a*d*d-b*b*e+2*b*c*d-c*c*c)
+(a*a*d-a*b*c-a*b*c+b*b*b)*(a*a*d-a*b*c-a*b*c+b*b*b)
-(a*c-b*b)*(a*a*a*e-a*a*b*d-a*a*b*d-a*a*c*c+3*a*b*b*c-b*b*b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ MR1999202 Zhang, Yusen + Wang, Tianming
\\ "Note on summation formulas derived from an identity of F. H. Jackson",
\\ Australas. J. Combin. 28 (2003), 295-304. (p. 297, equ. (7)):
\\ G. E. Andrews, R. P. Lewis, "An algebraic identity of F. H. Jackson and its
\\ implications for partitions", Disc. Math. 232 (2001) 77-83. (p. 78, equ.
\\ (1.7)): "We begin with a surprising but little known identity due to Jackson
\\ [4]: $$ 1 -\frac{a(1-b)(1-c)(1-d)(1-a^2bcd)}{(1-ab)(1-ac)(1-ad)(1-abcd)} =
\\ \frac{(1-a)(1-abc)(1-abd)(1-acd)}{(1-ab)(1-ac)(1-ad)(1-abcd)}. $$"
\\ {} 5 vars with 3 terms highest degree 5 of total 10
{ id5_3_5_10 = +a*e*(b-e)*(c-e)*(d-e)*(a*a*b*c*d-e*e*e*e*e)
+(a-e)*(a*b*c-e*e*e)*(a*b*d-e*e*e)*(a*c*d-e*e*e)
-(a*b-e*e)*(a*c-e*e)*(a*d-e*e)*(a*b*c*d-e*e*e*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 2 [NC]
{ id5_4_1_2a = +(a-e)*(b-e) -(c-e)*(d-e) -(a-e)*(b-d) -(a-c)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 2 [NC]
{ id5_4_1_2b = +(a-b)*(c-d) +(b-e)*(c-d) +(a-e)*(d-e) -(a-e)*(c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 2 [NC]
{ id5_4_1_2c = +(a-e)*(c-d) -(b-e)*(c-d) -(a-b)*(c-e) +(a-b)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 2 [TS]
{ id5_4_1_2d = +(a-b)*(e+b-c) +(b-c)*(e-a+b) +(c-d)*(e-a+d) -(a-d)*(e-c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 17, following equation):
\\ "(a+k)(b+k)(-n-1+k) -k(c+k-1)(-n+k+a+b-c) = -((c+n)(a+b-c+1)k +ab(n-k+1))"
\\ Replace {1->d}, then {a->c-d, b->e-b, c->-a+c-d+e, d->-a, n->d-e, k->d-e}.
\\ {} 5 vars with 4 terms highest degree 1 of total 3 [TS]
{ id5_4_1_3a = +a*(b-d)*(c-e) -a*(b-c)*(d-e) -b*(a-e)*(c-d) +e*(a-b)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011, (p. 18, Remark).
\\ {} 5 vars with 4 terms highest degree 1 of total 3 [TS]
{ id5_4_1_3b = +(a-c)*(a+c)*(b-d) -(a-e)*(a+e)*(b-d)
+(b-e)*(c+d)*(c-e) -(b+c)*(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ G. Chang and T. W. Sederberg, "Over and Over Again", MAA, 1997. (p. 288):
\\ "13.6 Hint: Verify the identity $$ (B-C)(x-A)(y-A)+(C-A)(x-B)(y-B)+
\\ (A-B)(x-C)(y-C)=(B-A)(C-B)(A-C). $$"
\\ {} 5 vars with 4 terms highest degree 1 of total 3
{ id5_4_1_3c = +(a-b)*(a-c)*(b-c) -(a-b)*(c-d)*(c-e)
+(a-c)*(b-d)*(b-e) -(a-d)*(a-e)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 3
{ id5_4_1_3d = +(a-b)*(b-c)*(c-d) +(a-b)*(c-d)*(c-e)
-(a-c)*(b-d)*(b-e) +(a-d)*(b-c)*(b-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 3 [TS]
{ id5_4_1_3e = +a*a*(c-d) +b*(d-e)*(b-c+e) -b*(c-e)*(b-d+e)
-(a-b)*(a+b)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 3
{ id5_4_1_3f = +(a-b)*(a-c)*(a-d) -(a-b)*(a-d)*(a-e)
+(a-c)*(a-d)*(c-e) -(a-d)*(b-c)*(c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 4 [TS]
{ id5_4_1_4a = +a*a*(b-d)*(c-e) -a*c*(a-e)*(b-d)
+b*e*(a-c)*(a-d) -d*e*(a-b)*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 4
{ id5_4_1_4b = +(a-b)*(c-d)*(c-e)*(d-e) -(a-c)*(b-d)*(b-e)*(d-e)
+(a-d)*(b-c)*(b-e)*(c-e) -(a-e)*(b-c)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 4 [TS]
{ id5_4_1_4c = +(a-b)*(a+b)*(c-d)*(b-e) -(a-c)*(a+c)*(b-d)*(c-e)
+(a-d)*(a+d)*(b-c)*(d-e) +(b-c)*(b-d)*(c-d)*(b+c+d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 4 [TS]
{ id5_4_1_4d = +(a-b)*(a+b)*(c-d)*(c-e) -(a-c)*(a+c)*(b-d)*(b-e)
+(a-d)*(a-e)*(b-c)*(b+c) -(a-b)*(a-c)*(b-c)*(d+e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 5 [TS]
{ id5_4_1_5a = +a*a*(b-c)*(c-d)*(b+e) -a*a*(b-c)*(c-e)*(b+d)
+b*b*(a-c)*(d-e)*(a+c) -c*c*(a-b)*(d-e)*(a+b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 1 of total 5 [TS]
{ id5_4_1_5b = +a*b*(c-d)*(c-e)*(d-e) -c*d*(a-e)*(b-e)*(c-d)
+c*e*(a-d)*(b-d)*(c-e) -d*e*(a-c)*(b-c)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Max Alekseyev, Math Overflow question 280195, Sep 02 2017 case n=2:
\\ $$ (k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n
\\ \frac{t-d_j}{d_i-d_j}, $$
\\ {} 5 vars with 4 terms highest degree 1 of total 5 [TS]
{ id5_4_1_5c = +(a-e)*(a-e)*(b-c)*(b-d)*(c-d)
-(b-e)*(b-e)*(a-c)*(a-d)*(c-d)
+(c-e)*(c-e)*(a-b)*(a-d)*(b-d)
-(d-e)*(d-e)*(a-b)*(b-c)*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into four rectangles.
\\ {} 5 vars with 4 terms highest degree 2 of total 2 [NC]
{ id5_4_2_2 = +(a*b-c*d) -a*(b-e) +c*(d-e) -(a-c)*e ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 2 of total 3
{ id5_4_2_3a = +a*c*c -b*b*d +b*(b*d+c*e) -c*(a*c+b*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 2 of total 3
{ id5_4_2_3b = +a*(b-d)*(c-e) +a*(c-e)*(d-e) -e*(a-b)*(a-c) +(a-e)*(a*e-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 2 of total 4
{ id5_4_2_4a = +a*a*(a*a-b*c) -a*b*d*(a-e)
-b*d*(a*e-b*c) -(a*a-b*c)*(a*a-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 2 of total 4
{ id5_4_2_4b = +c*(a-e)*(b-d)*(d-e) +d*(a-c)*(b-e)*(c-e)
-(b-d)*(c-e)*(a*d-c*e) -(a-c)*(d-e)*(b*c-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 2 of total 5
{ id5_4_2_5a = +a*b*(c-d)*(c-e)*(d-e) +d*e*(a-b)*(a-c)*(b-c)
-a*(b-d)*(c-e)*(b*d-c*e) -d*(a-e)*(b-c)*(a*e-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 4 terms highest degree 2 of total 5
{ id5_4_2_5b = +b*d*(a-e)*(c-e)*(c-e) -c*e*(a-d)*(b-d)*(b-e)
-b*(c-e)*(d-e)*(a*c-d*e) +e*(a-d)*(b-c)*(b*c-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Tom H. Koornwinder, "On the equivalence of two fundamental theta identities",
\\ arXiv:1401.5368v2, page 4 equation (2.8)
\\ {} 5 vars with 4 terms highest degree 3 of total 12
{ id5_4_3_12 = +a*b*c*d*(a*a-e*e)*(b*b-e*e)*(c*c-e*e)*(d*d-e*e)
+a*b*c*d*(a*a+e*e)*(b*b+e*e)*(c*c+e*e)*(d*d+e*e)
+(a*b*c-d*e*e)*(a*b*d-c*e*e)*(a*c*d-b*e*e)*(a*e*e-b*c*d)
-(a*b*c+d*e*e)*(a*b*d+c*e*e)*(a*c*d+b*e*e)*(a*e*e+b*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This appears in my proof of a lemniscate sine addition theorem Jun 13 2023
\\ at "https://math.stackexchange.com/q/4587359". Replace {sl(x) -> a/b,
\\ sl(y) -> c/b, cl(x) -> d/b, cl(y) -> e/b}.
\\ {} 5 vars with 4 terms highest degree 4 of total 8
{ id5_4_4_8 = +(a*c*d*e-b*b*b*b)*(a*a*c*d+a*b*b*e+a*c*c*e+b*b*c*d)
+a*a*c*d*(b*b*b*b-b*b*c*c-b*b*e*e-c*c*e*e)
-a*c*c*e*(a*a*b*b+a*a*d*d-b*b*b*b+b*b*d*d)
+(a*e+c*d)*(a*a*c*c+b*b*b*b)*b*b ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ R. Langer, M. J. Schlosser, S. Ole Warnaar, "Theta Functions, Elliptic
\\ Hypergeometric Series, and Kawanaka's Macdonald Polynomial Conjecture"
\\ arXiv:0905.4033v1, page 3 equation (1.4)
\\ {} 5 vars with 4 terms highest degree 5 of total 15
{ id5_4_5_15 =
+(a-b)*(a*a-c*d)*(a*a-c*e)*(a*a-d*e)*(a*a*a-b*d*e)*(a*a*a*a*a-b*c*d*d*e)
+(a-c)*(a*a-b*d)*(a-e)*(a*a*a-b*d*e)*(a*a*a*a-b*c*d*e)*(a*a*a*a-c*d*d*e)
-(a-b)*(a*a-c*d)*(a-e)*(a*a*a-c*d*e)*(a*a*a*a-c*b*d*e)*(a*a*a*a-b*d*d*e)
-(a-c)*(a*a-b*d)*(a*a-b*e)*(a*a-d*e)*(a*a*a-c*d*e)*(a*a*a*a*a-c*b*d*d*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id5_4_2_2 with its first term split into two terms.
\\ {} 5 vars with 5 terms highest degree 1 of total 2 [NC]
{ id5_5_1_2a = +a*b -c*d -a*(b-e) +c*(d-e) -(a-c)*e ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 2 [NC]
{ id5_5_1_2b = +(a-b)*(c-d) -(b-c)*(d-e) -a*(c-d) +b*(c-e) -c*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 2 [NC]
{ id5_5_1_2c = +(a-c)*(b-d) +(a-c)*(d-e) -(a-e)*(b-e)
+(c-e)*(b-d) +(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 2 [NC]
{ id5_5_1_2d = -(a-e)*(b-e) +(a-c)*(b-c) +(a-c)*(c-e)
+(c-e)*(b-d) +(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 2 [NC]
{ id5_5_1_2e = +(a-b)*(a-c) -(a-e)*(a-c) +(b-d)*(a-e)
+(d-e)*(a-e) -(b-e)*(c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 2 [NC]
{ id5_5_1_2f = +(a-b)*(b-c) -(a-e)*(b-d) +(a-e)*(c-e)
-(a-e)*(d-e) +(b-e)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ If a+b = c+d then a*b - c*d = (a+x)*(b+x) - (c+x)*(d+x) for all x since
\\ (a+x)*(b+x) = a*b + (a+b)*x + x^2.
\\ {} 5 vars with 5 terms highest degree 1 of total 2 [TS]
{ id5_5_1_2g = +a*b -c*d -(a+e)*(b+e) +(c+e)*(d+e) +e*(a+b-c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ B. C. Berndt, Ramanujan's Notebooks, Part IV, (p. 36., (25.1)) case n=3 :
\\ "$$ \sum_{k=0}^{n-1} \frac{a_0a_1\dots a_k}{(x+a_1)(x+a_2)\dots(x+a_{k+1})}
\\ = \frac{1}{x}-\frac{1}{x(1+x/a_1)(1+x/a_2)\dots(1+x/a_n)}. $$"
\\ {} 5 vars with 5 terms highest degree 1 of total 3 [NC]
{ id5_5_1_3a = +(a-b)*(a-c)*(a-d) -(a-e)*(a-c)*(a-d) +(b-e)*(a-e)*(a-d)
-(b-e)*(c-e)*(a-e) +(b-e)*(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 3
{ id5_5_1_3b = +(a-e)*(b-e)*(c-e) -(a-e)*(b-e)*(d-e) -(a-d)*(b-e)*(c-e)
+(a-c)*(b-c)*(d-e) +(a-c)*(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 3
{ id5_5_1_3c = +(a-d)*(b-e)*(c-e) -(a-e)*(b-e)*(c-e) +(b-e)*(b-e)*(d-e)
-(b-c)*(b-c)*(d-e) -(b-c)*(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 5 terms highest degree 1 of total 3
{ id5_5_1_3d = +(a-d)*(b-e)*(c-e) -(a-e)*(b-e)*(c-e) +(b-e)*(d-e)*(d-e)
+(b-c)*(c-d)*(d-e) +(c-d)*(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ H. S. Hall and S. R. Knight, Higher Algebra, 1957 (p. 257, Examples 12(2)).
\\ T. Amdeberhan, O. Espinosa, V. Moll, A. Straub, "Wallis-Ramanujan-Schur-
\\ Feynman", (p.5, equ. (4.3)) case n=4 : "\prod_{k=1}^n \frac{1}{y+b_k} =
\\ \sum_{k=1}^n \frac{1}{y+b_k}\prod_{j=1,j\ne k}^n\frac{1}{b_j-b_k}. "
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=5, r=0 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 5 vars with 5 terms highest degree 1 of total 6
{ id5_5_1_6 = +(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d)
-(a-b)*(a-c)*(a-e)*(b-c)*(b-e)*(c-e)
+(a-b)*(a-d)*(a-e)*(b-d)*(b-e)*(d-e)
-(a-c)*(a-d)*(a-e)*(c-d)*(c-e)*(d-e)
+(b-c)*(b-d)*(b-e)*(c-d)*(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=5, r=1 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 5 vars with 5 terms highest degree 1 of total 7 [TS]
{ id5_5_1_7 = +a*(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)
-b*(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)
+c*(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)
-d*(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)
+e*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=5, r=2 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 5 vars with 5 terms highest degree 1 of total 8
{ id5_5_1_8 = +a*a*(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)
-b*b*(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)
+c*c*(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)
-d*d*(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)
+e*e*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 226) :
\\ "Leonardo Pisano,^3 in his Liber Quadratorum of 1225, proved (1) and used
\\ it to solve $x^2+y^2 = a^2+b^2,$ given a solution of $c^2+d^2=e^2:$
\\ $$ x=(ac+bd)/e, \quad y=(ad-bc)/e. $$
\\ {} 5 vars with 5 terms highest degree 2 of total 4
{ id5_5_2_4 = +(a*c+b*d)*(a*c+b*d) +(a*d-b*c)*(a*d-b*c) -a*a*e*e -b*b*e*e
-(a*a+b*b)*(c*c+d*d-e*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Show that $ 1/(x+a)-1/(x+b) = (1/(1-x*a)-1/(1-x*b))/x $ if $ ab = -1 $
\\ From "https://math.stackexchange.com/q/2359755" asked by Leyla Alkan.
\\ {} 5 vars with 5 terms highest degree 2 of total 7
{ id5_5_2_7 = +a*(a*c+b*d)*(a*d-b*c)*(a*e-b*c)
-a*(a*c+b*e)*(a*d-b*c)*(a*e-b*c)
+b*(a*c+b*d)*(a*c+b*e)*(a*d-b*c)
-b*(a*c+b*d)*(a*c+b*e)*(a*e-b*c)
-a*b*(d-e)*(a*a+b*b)*(c*c+d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 6 terms highest degree 1 of total 2 [NC]
{ id5_6_1_2a = +a*a -b*c -(a-d)*(a-e) +(b-d)*(c-e) -(a-b)*e -d*(a-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 6 terms highest degree 1 of total 2 [NC]
{ id5_6_1_2b = +a*d +b*e -(a+b)*(d+e) -(a+c)*d -(b+c)*e +(a+b+c)*(d+e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the determinant of the 3x3 rank 2 matrix
\\ [a+b+c-d, a-d, a+c+e; a+b-e, a-c-e, a+d; a, a-b-c, a-b+d+e]
\\ {} 5 vars with 6 terms highest degree 1 of total 3 [TS]
{ id5_6_1_3 = +(a+b+c-d)*(a-c-e)*(a-b+d+e) +a*(a+d)*(a-d)
+(a-b-c)*(a+c+e)*(a+b-e) -(a+b+c-d)*(a+d)*(a-b-c)
-a*(a-c-e)*(a+c+e) -(a-b+d+e)*(a-d)*(a+b-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 6 terms highest degree 1 of total 5 [JE]
{ id5_6_1_5a = +(a-e)*(c-e)*(d-e)*(a-c)*(a-d) -(b-e)*(c-e)*(d-e)*(a-d)*(b-c)
-(a-e)*(b-e)*(b-e)*(a-b)*(a-b) -(b-e)*(b-e)*(d-e)*(a-b)*(b-d)
+(b-e)*(a-b)*(a-b)*(a-d)*(b-d) +(d-e)*(a-b)*(a-c)*(a-d)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 6 terms highest degree 1 of total 5 [JE]
{ id5_6_1_5b = +(a-e)*(b-e)*(d-e)*(a-b)*(a-d) -(a-e)*(b-e)*(c-e)*(a-b)*(a-c)
-(b-e)*(c-e)*(d-e)*(a-c)*(b-d) +(b-e)*(c-e)*(d-e)*(a-d)*(b-c)
+(c-e)*(a-b)*(a-c)*(a-d)*(b-d) -(d-e)*(a-b)*(a-c)*(a-d)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 6 terms highest degree 1 of total 10
{ id5_6_1_10 = +a*b*c*d*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)
-a*b*c*e*(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)
+a*b*d*e*(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)
-a*c*d*e*(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)
+b*c*d*e*(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)
-(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ See https://math.stackexchange.com/q/433577/
\\ {} 5 vars with 6 terms highest degree 2 of total 8
{ id5_6_2_8 = +a*a*a*a*a*a*(b-c)*(d-e) -a*a*b*c*d*e*(b-c)*(d-e)
-b*d*(a*a-b*e)*(a*a-c*d)*(a*a-c*e)
+b*e*(a*a-b*d)*(a*a-c*d)*(a*a-c*e)
+c*d*(a*a-b*d)*(a*a-b*e)*(a*a-c*e)
-c*e*(a*a-b*d)*(a*a-b*e)*(a*a-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 6 terms highest degree 4 of total 5
{ id5_6_4_5 = +(a*a-b*c)*(a*a*d+b*c*e) +(b*b-a*c)*(b*b*d+a*c*e)
+(c*c-a*b)*(c*c*d+a*b*e) +e*(a*a*b*b+b*b*c*c+a*a*c*c)
+a*b*c*(d-e)*(a+b+c) -d*(a*a*a*a+b*b*b*b+c*c*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 6 terms highest degree 4 of total 9
{ id5_6_4_9 = +a*e*(b-e)*(c-e)*(d-e)*(a*b*c*d+e*e*e*e)
+a*b*e*e*(c-e)*(d-e)*(a*c*d-e*e*e)
+a*c*e*e*(b-e)*(d-e)*(a*b*d-e*e*e)
+a*d*e*e*(b-e)*(c-e)*(a*b*c-e*e*e)
+(a-e)*(a*b*c*d-e*e*e*e)*(a*b*c*d-e*e*e*e)
-(a*b*c-e*e*e)*(a*b*d-e*e*e)*(a*c*d-e*e*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 5 vars with 7 terms highest degree 1 of total 4
{ id5_7_1_4 = +a*b*c*(a-b) -a*c*d*(a-d) +b*c*(b-d)*(b-c+d) -b*e*(b-d)*(b-e)
-c*(a-b)*(a-d)*(b-d) +c*e*(b-d)*(c-e) -(b-c)*(b-d)*(b-e)*(c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 6 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 2): "Now consider
\\ (1-aq^k)b - (1-bq^k)a = b-a. ..."
\\ T. Amdeberhan, O. Espinosa, V. Moll, A. Straub, "Wallis-Ramanujan-Schur-
\\ Feynman", (p.5, equ. (4.3)) case n=2 : "\prod_{k=1}^n \frac{1}{y+b_k} =
\\ \sum_{k=1}^n \frac{1}{y+b_k}\prod_{j=1,j\ne k}^n\frac{1}{b_j-b_k}. "
\\ English Wikipedia article https://en.wikipedia.org/wiki/Linear_relation
\\ "Cayley used it to denote linear relations between minors of a matrix,
\\ such as, in the case of a 2x3 matrix:
\\ a|b c;e f| - b|a d;d f| + c|a b;d e| = 0."
\\ Let A = 1/(1-x*y), B = 1/(1-x*z). Then A - B = A*B*x*(y-z). Replace
\\ {x -> a/f, y -> e/b, z -> c/d} to get this identity.
\\ {} 6 vars with 3 terms highest degree 2 of total 3
{ id6_3_2_3a = +a*(b*c-d*e) -b*(a*c-d*f) +d*(a*e-b*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "A Short Proof of an Identity of Sylvester", Internat. J.
\\ Math. & Math. Sci., v. 22, n. 2 (1999) 431-435, (p. 432, equ. 2.1) case n=2
\\ of an identity due to Milne: \sum_{r=1}^n (1-y_r) \prod_{i=1,i\ne r}^n
\\ [\frac{1-x_iy_i/x_r}{1-x_i/x_r}] = 1-y_1y_2\cdots y_n.
\\ George E. Andrews, "q-Series: ...", 1986 AMS, (p. 111) : " [[...]]
\\ ((1-bq^n)(1-c)-(1-cq^n)(1-b)) [[...]] (b-c)(1-q^n) [[...]]".
\\ Jens Carsten Jantzen, Lectures on Quantum Groups, AMS, 1996. p. 13: "[[...]]
\\ (q^b-q^{-b})(Kq^{a+c}-K^{-1}q^{-a-c})+(q^c-q^{-c})(Kq^{a-b}-K^{-1}q^{-a+b})
\\ = [[...]] = (q^{b+c}-q^{-b-c})(Kq^a-K^{-1}q^{-a}), [[...]]"
\\ {} 6 vars with 3 terms highest degree 2 of total 3
{ id6_3_2_3b = +(a-d)*(b*f-c*e) -(b-e)*(a*f-c*d) +(c-f)*(a*e-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 3 terms highest degree 2 of total 3
{ id6_3_2_3c = +a*(a*b-a*c+b*d-c*f) -b*(a*a+a*d-c*e) +c*(a*a-b*e+a*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Given recursion $u_{n+1} = tu_n - u_{n-1}$ prove $u_{n-1}u_{n+1} - u_n^2$
\\ is a constant sequence by induction using this identity. Replace {a -> 1,
\\ b -> t, c -> u_{n-1}, d -> u_n, e -> u_{n+1}, f -> u_{n+2}}.
\\ {} 6 vars with 3 terms highest degree 2 of total 3
{ id6_3_2_3d = +a*(d*d-c*e+d*f-e*e) -d*(a*d-b*e+a*f) +e*(a*c-b*d+a*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 107) : "Guha's
\\ Inequality. If $p\ge q>0, x\ge y>0$ then $(px+y+a)(x+qy+a)\ge[(p+1)x+a]
\\ [(q+1)y+a].$ [[...]] the difference [[...]] is just $(px-qy)(x-y)$."
\\ {} 6 vars with 3 terms highest degree 2 of total 4
{ id6_3_2_4a = +a*(b-c)*(b*e-c*f) +(a*b+a*d+b*e)*(a*c+a*d+c*f)
-(a*b+a*d+c*f)*(a*c+a*d+b*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 3 terms highest degree 2 of total 4
{ id6_3_2_4b = +(a*c-b*b)*(d*e-f*f) +(a*d-b*f)*(b*f-c*e)
-(a*f-b*e)*(b*d-c*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Let $ U_n := (a^n-b^n)/(a-b) $ the first Lucas sequence and $ Q := ab $.
\\ Replace {a^n -> a, b^n -> b, a^m -> c, b^m -> d, a^k -> e, b^k -> f} in
\\ the identity $0 = Q^nU_{m+n}U_{m-n} - Q^nU_{m+k}U_{m-k} +
\\ Q^mU_{n+k}U_{n-k}$
\\ {} 6 vars with 3 terms highest degree 2 of total 6
{ id6_3_2_6a = +a*b*(c*e-d*f)*(c*f-d*e) -c*d*(a*e-b*f)*(a*f-b*e)
+e*f*(a*c-b*d)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Let $ U_n := (a^n-b^n)/(a-b) $ the first Lucas sequence and $ Q := ab $.
\\ Replace {a^n -> a, b^n -> b, a^m -> c, b^m -> d, a^k -> e, b^k -> f} in
\\ the identity $ 0 = U_k^2U_{m+n}U_{m-n}Q^n - U_n^2U_{m+k}U_{m-k}Q^k +
\\ U_m^2U_{n+k}U_{n-k}Q^k $
\\ {} 6 vars with 3 terms highest degree 2 of total 6
{ id6_3_2_6b = +(a-b)*(a-b)*(c*e-d*f)*(c*f-d*e)
-(c-d)*(c-d)*(a*e-b*f)*(a*f-b*e)
+(e-f)*(e-f)*(a*c-b*d)*(a*d-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The Pentagon, v. 67, n. 1, Fall 2007, pp. 46-47 "Problem 600. Proposed by
\\ Stanley Rabinowitz, MathPro Press". p. 47: "[[...]]] if and only if
\\ $$(xy+y+1)(yz+z+1)(zx+x+1) = (xy+x+1)(yz+y+1)(zx+z+1).$$ A little algebra
\\ shows that this condition is equivalent to $(x-y)(y-z)(z-x)=0.$"
\\ {} 6 vars with 3 terms highest degree 2 of total 6
{ id6_3_2_6c = +(a*b+b*d+d*e)*(a*c+a*f+d*f)*(b*c+e*c+e*f)
-(a*b+a*e+d*e)*(a*c+c*d+d*f)*(b*c+b*f+e*f)
-(a*e-b*d)*(a*f-c*d)*(b*f-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 2): "Next consider
\\ (1-aq^k) - (1-bq^k) = q^k(b-a). ..."
\\ {} 6 vars with 3 terms highest degree 3 of total 3
{ id6_3_3_3a = +(a*c*e-b*c*d) +d*(b*c-e*f) -e*(a*c-d*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 3 terms highest degree 3 of total 3
{ id6_3_3_3b = +(a*b*f-c*d*e) -a*f*(b-c) -c*(a*f-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "A Short Proof of an Identity of Sylvester", Internat. J.
\\ Math. & Math. Sci., v. 22, n. 2 (1999) 431-435, (p. 432, equ. 2.1) case n=2
\\ of an identity due to Milne: \sum_{r=1}^n (1-y_r) \prod_{i=1,i\ne r}^n
\\ [\frac{1-x_iy_i/x_r}{1-x_i/x_r}] = 1-y_1y_2\cdots y_n.
\\ {} 6 vars with 3 terms highest degree 3 of total 4
{ id6_3_3_4a = +(a*f-c*d)*(b*e-c*f) -(b-c)*(a*f*f-c*d*e)
-(e-f)*(a*b*f-c*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 3): "... is equivalent with
\\ (1-C)(1-C/AB) + C/AB(1-A)(1-B) = (1-C/A)(1-C/B). (3.1) ... After some
\\ simplification (3.1) reduces to (1-aq^k)(1-b)-(1-a)(1-bq^k)=(a-b)(1-q^k)."
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011, (p. 31): "[[...]] or
\\ (1-b)(1-c)a -(1-a)(a-bc) = (a-b)(a-c)."
\\ Gasper+Rahman, Basic Hypergeometric Series 2nd ed. (p. 17, equ. (1.7.2))
\\ case n=1 of q-Pfaff-Saalschutz formula
\\ _3_\phi_2(a,b,q^{-n};c,abq/cq^n;q;q) = \frac{(c/a)_n(c/b)_n}{(c)_n(c/ab)_n}
\\ Let S_n := x^n-y^n. It satisfies the identity $$ S_nS_{n+i-j} -
\\ S_{n+i}S_{n-j} = (xy)^(n-j)S_iS_j. $$ Replace {x^n -> a, y^n -> b,
\\ x^j -> c, y^i -> d, y^j ->e, x^i -> f}.
\\ {} 6 vars with 3 terms highest degree 3 of total 4
{ id6_3_3_4b = +a*b*(c-e)*(d-f) -(a*e-b*c)*(a*f-b*d) +(a-b)*(a*e*f-b*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 3 terms highest degree 3 of total 4
{ id6_3_3_4c = +a*(a*b*c+b*d*e-a*e*e-b*b*f) -b*(a*a*c+b*d*d-a*b*f-a*d*e)
+(a*e-b*d)*(a*e-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Hans Reisel, Prime Numbers and Computer Methods for Factorization, 2nd ed.,
\\ BirkHauser Boston, 1994. p. 263 : "ac = (b+kn)(d+ln) = bd+(kd+bl+kln)n,"
\\ {} 6 vars with 3 terms highest degree 3 of total 4
{ id6_3_3_4d = +a*a*b*c +e*(a*b*f+a*c*d+d*e*f) -(a*b+d*e)*(a*c+e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From Wikipedia, "In algebra, Brahmagupta's identity says that, for given
\\ n, the product of two numbers of the form a^2+nb^2 is itself a number of
\\ that form." This is a homogeneous form of the Brahmagupta Identity.
\\ Albert H. Beiler, Recreations in the Theory of Numbers, Dover Publications,
\\ 1964, p. 232.
\\ {} 6 vars with 3 terms highest degree 3 of total 6
{ id6_3_3_6 = +(e*a*a+f*b*b)*(e*c*c+f*d*d) -(e*a*c-f*b*d)*(e*a*c-f*b*d)
-e*f*(a*d+b*c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ MR1999202 Zhang, Yusen + Wang, Tianming
\\ "Note on summation formulas derived from an identity of F. H. Jackson",
\\ Australas. J. Combin. 28 (2003), 295-304. (p. 295, equ. (1)) case n=1.
\\ {} 6 vars with 3 terms highest degree 3 of total 8
{ id6_3_3_8 = +(a-d)*(a*b-c*d)*(a*f-c*e)*(a*a*a-b*e*f)
-(a-f)*(a*c-b*f)*(a*d-b*e)*(a*a*a-c*d*e)
-(a*a-b*e)*(a*a-c*e)*(a*a-d*f)*(b*f-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 3 terms highest degree 4 of total 5
{ id6_3_4_5a = +a*b*(b*d*d-c*c*e) +c*e*(a*b*c-d*e*f) -d*(a*b*b*d-c*e*e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 3 terms highest degree 4 of total 5
{ id6_3_4_5b = +a*a*(b*c*f-a*d*e) -b*f*(a*a*c-b*b*d) +d*(a*a*a*e-b*b*b*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Wilhelm Magnus, Noneuclidean Tesselations and Their Groups, Academic Press,
\\ Inc., 1974, page 4, "$\alpha\delta-\beta\gamma = 1,$" and the following
\\ equations (1.11a) -- (1.11e).
\\ {} 6 vars with 3 terms highest degree 4 of total 6
{ id6_3_4_6 =
+(a*c*d*e+a*b*e*f-a*b*d*f-a*c*e*f-b*c*d*e+b*c*d*f)*(a*e-a*f-b*d+b*f+c*d-c*e)
+(a*b*d-a*b*e-a*c*d+a*c*f+b*c*e-b*c*f)*(a*d*e-a*d*f-b*d*e+b*e*f+c*d*f-c*e*f)
-(a-b)*(a-c)*(b-c)*(d-e)*(d-f)*(e-f)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 591) :
\\ "From the fact that (7) satisfy Ax^3+By^3+Cz^3=0, we have the identity
\\ (b^3c'^3-b'^3c^3)(a^2b'c'-a'^2bc)^3 + (c^3a'^3-c'^3a^3)(b^2a'c'-b'^2ac)^3
\\ + (a^3b'^3-a'^3b^3)(c^2a'b'-c'^2ab)^3 = 0."
\\ Now let d=a', e=b', f=c', and factor x^3-y^3 = (x-y)*(x*x+x*y+y*y).
\\ {} 6 vars with 3 terms highest degree 4 of total 18
{ id6_3_4_18 =
+(b*f-c*e)*(b*b*f*f+b*c*e*f+c*c*e*e)*(a*a*e*f-d*d*b*c)*(a*a*e*f-d*d*b*c)*(a*a*e*f-d*d*b*c)
+(c*d-a*f)*(c*c*d*d+c*d*a*f+a*a*f*f)*(b*b*d*f-e*e*a*c)*(b*b*d*f-e*e*a*c)*(b*b*d*f-e*e*a*c)
+(a*e-b*d)*(a*a*e*e+a*b*d*e+b*b*d*d)*(c*c*d*e-f*f*a*b)*(c*c*d*e-f*f*a*b)*(c*c*d*e-f*f*a*b)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 3 terms highest degree 5 of total 10
{ id6_3_5_10 = +a*c*(b-f)*(d-f)*(e-f)*(a*a*f*f*f-b*c*c*d*e)
-(a-c)*(a*f*f-b*c*d)*(a*f*f-b*c*e)*(a*f*f-c*d*e)
+(a*f-b*c)*(a*f-c*d)*(a*f-c*e)*(a*f*f*f-b*c*d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into three rectangles.
\\ case n=2 of 0 = (x_1-y_1)(x_2-y_2)...(x_n-y_n) +
\\ (y_1-z_1)*(x_2-y_2)...(x_n-y_n) + (x_1-z_1)*(y_2-z_2)*(x_3-y_3)...(x_n-y_n)
\\ + ... + (x_1-z_1)*(x_2-z_2)...(y_n-z_n) - (x_1-z_1)*(x_2-z_2)...(x_n-z_n).
\\ N. J. Wildberger, Divine Proportions, 2005, (p. 27, Exer. 2.6) : "Check the
\\ following identity, [[...]] $$ (y_1-y_2)(x_3-x_1)-(y_1-y_3)(x_2-x_1) =
\\ (y_1-y_3)(x_3-x_2)-(y_2-y_3)(x_3-x_1). $$"
\\ {} 6 vars with 4 terms highest degree 1 of total 2 [NC]
{ id6_4_1_2a = +(a-b)*(d-e) +(b-c)*(d-e) +(a-c)*(e-f) -(a-c)*(d-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 1 of total 2 [TS]
{ id6_4_1_2b = +(a+b-d)*(c-d+f) -(a-b-e)*(c+e-f) -(a+c-d)*(b-d+f)
+(a-c-e)*(b+e-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 18, Remark):
\\ (c+n)(-n+k+a+b-c)(n+1)-(c-a+n)(c-b+n)(-n+k-1)=(c+n)(a+b-c+1)k +ab(n-k+1)
\\ {} 6 vars with 4 terms highest degree 1 of total 3 [TS]
{ id6_4_1_3 = +(a-b)*(c-d)*(e-f) +(a-c)*(b-f)*(d-e)
-(a-b)*(c-e)*(d-f) -(a-f)*(b-c)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 1 of total 4 [TS]
{ id6_4_1_4a = +(a-b)*(a-c)*(d-f)*(e-f) -(a-c)*(a-d)*(b-f)*(e-f)
+(a-e)*(a-f)*(b-d)*(c-f) -(a-f)*(a-f)*(b-d)*(c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Shaun Cooper and Michael Hirschhorn, "On Some Infinite Product Identities",
\\ Rocky Mountain J. Math. 31 (2001), no. 1, 131-139, (p. 132 Lemma) case n=3:
\\ "[[...]] $$ [a;q]_\infty = (a;q)\infty(a^{{-1}q;q)\infty $$ [[...]] Lemma.
\\ Suppose $ a_1,a_2,\dots,a_n; b_1,b_2,\dots,b_n $ are nonzero complex numbers
\\ which satisfy (i) $a_i\neq q^na_j$ for all $i\neq j$ and all $n\in Z$, (ii)
\\ $a_1a_2\cdots a_n=b_1b_2\cdots b_n$. Then $$ \sum_{i=1}^n \frac{ \prod_{j=1}
\\ ^n[a_ib_j^{-1};q]_\infty}{\prod_{j=1,j\neq i}^n[a_ib_j^{-1};q]_\infty}}=0. $$"
\\ Replace [x y^{-1};q]_\infty with (x-y) throughout. Replace
\\ {a_1 -> a, a_2 -> b, a_3 -> c, b_1 -> d, b_2 -> e, b_3 -> f}.
\\ Arthur Cayley, Theorem in Trigonometry, Messenger of Math., 5 (1876),
\\ p. 164 : "If $A + B + C + F + G + H = 0$, then $| sin A+F sin B+F sin C+F,
\\ cos F, sin F & sin A+G sin B+G sin C+G, cos G, sin G & sin A+H sin B+H sin
\\ C+H, cos H, sin H | = 0.$".
\\ {} 6 vars with 4 terms highest degree 1 of total 4 [TS]
{ id6_4_1_4b = +(a-b)*(c-d)*(c-e)*(c-f) -(a-c)*(b-d)*(b-e)*(b-f)
+(a-d)*(a-e)*(a-f)*(b-c) -(a-b)*(a-c)*(b-c)*(a+b+c-d-e-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 1 of total 4 [TS]
{ id6_4_1_4c = +a*b*c*(c-e-f) -c*d*(a+b-d)*(c-e-f) +e*f*(a-d)*(b-d)
-(a-d)*(b-d)*(c-e)*(c-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Wenchang Chu and Ying You, "Binomial Symmetries inspired by Bruckman's
\\ Problem", Filomat 24:1 (2010), 41-46, (p. 42 Theorem 1) case n=2:
\\ [[...]] Let x and y be two indeterminate and {\alpha_k,\gamma_k}_{k=1}^n
\\ complex numbers with {\gamma_k}_{k=1}^n being distinct. Then there hold
\\ the algebraic identity $$ \sum_{k=1}^n \frac{(\alpha_k-\gamma_k)(x-y)}
\\ {(x+\gamma_k)(y+\gamma_k)} \prod_{i=1, i\ne k}^n \frac{\alpha_i-\gamma_k}
\\ {\gamma_i-\gamma_k} = \prod_{i=1}^n\frac{y+\alpha_i}{y+\gamma_i}
\\ -\prod_{i=1}^n\frac{x+\alpha_i}{x+\gamma_i}. $$
\\ {} 6 vars with 4 terms highest degree 1 of total 5 [TS]
{ id6_4_1_5 = +(a-f)*(b-f)*(c-d)*(c-e)*(d-e) -(c-f)*(d-f)*(a-e)*(b-e)*(c-d)
+(c-f)*(e-f)*(a-d)*(b-d)*(c-e) -(d-f)*(e-f)*(a-c)*(b-c)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ case n=3 of identity ??? (I have lost the source reference to this)
\\ {} 6 vars with 4 terms highest degree 1 of total 7 [WS]
{ id6_4_1_7 = +(a-b)*(a-c)*(b-c)*(a+d)*(b+e)*(c+f)*(d+e+f)
-a*b*f*(a-b)*(a-c+d)*(b-c+e)*(c+d+e+f)
-a*c*e*(a-c)*(a-b+d)*(b-c-f)*(b+d+e+f)
-b*c*d*(b-c)*(a-b-e)*(a-c-f)*(a+d+e+f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 3
{ id6_4_2_3a = -b*(a*c-d*f) +b*d*(c-f) +c*e*(a-d) +c*(a-d)*(b-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ A. Berkovich and K. Grizzell, "On the Class of Dominant and Subordinate
\\ Products. (p. 2) case L=3 : "$$ (1.6) \frac{1}{P(L)} -\frac{1}{Q(L)} =
\\ \sum_{i=1}^L \\ \frac{Q(i-1)}{P(i)Q(L)} (\frac{Q(i)}{Q(i-1)} -\frac{P(i)}
\\ {P(i-1)}) $$
\\ {} 6 vars with 4 terms highest degree 2 of total 3
{ id6_4_2_3b = +a*b*(c-d) +a*(b*d-c*f) -b*c*(a-e) +c*(a*f-b*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 3
{ id6_4_2_3c = -(a-d)*(b*c-e*f) +(a-d)*(b*f-c*e)
+(c-f)*(a*b-d*e) +(c-f)*(a*e-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The Lyness 5-cycle: If a(n+2) = (a(n+1)+1)/a(n) for n=1,2,3,4, then
\\ a(6)=a(1) if all a(n) are non-zero. One proof uses the algebraic identity
\\ 0 = (a2*a4-a3-1) -(a3*a5-a4-1) +a4*(a1*a3-a2-1) -a3*(a4*a1-a5-1). Now
\\ replace {a1 -> e/a, a2 -> f/a, a3 -> b/a, a4 -> c/a, a5 -> d/a}.
\\ {} 6 vars with 4 terms highest degree 2 of total 3
{ id6_4_2_3d = +a*(a*a+a*b-c*f) -a*(a*a+a*c-b*d)
-b*(a*a+a*d-c*e) +c*(a*a-b*e+a*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 4
{ id6_4_2_4a = +a*(a-d)*(a*a-b*c) +a*(a-f)*(a*d-b*e)
-(a*a-b*c)*(a*a-b*e) +(a*d-b*e)*(a*f-b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 4
{ id6_4_2_4b = +a*c*e*(a-b) +a*b*(c-d)*(e-f)
-b*d*f*(a-b) -(a*c-b*d)*(a*e-b*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 4
{ id6_4_2_4c = +a*c*e*e +b*d*f*f -(a*e-b*f)*(c*e-d*f) -e*f*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 22): "We find that [[...]]
\\ (1-aq^k)(1-q^{-n-1+k})bq^n/a -(1-bq^{k-1})(1-q^k)
\\ = b/a(1-a)q^n(1-q^{-n+k-1}) -(1-bq^n)(1-q^k), [[...]]"
\\ {} 6 vars with 4 terms highest degree 2 of total 5
{ id6_4_2_5a = +c*e*(a-b)*(a*a-d*f) -a*c*(a-b)*(a*e-b*d)
-d*(a*a-b*c)*(a*b-e*f) +a*d*(a-c)*(a*b-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 5
{ id6_4_2_5b = +b*(a*f-c*d)*(b*c+e*f) -c*(a*b+d*e)*(b*f-c*e)
-c*(a*e-b*d)*(b*c+e*f) -e*(a*f-c*d)*(b*f-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 26, equ. (8.4)): "elementary relation:
\\ 1-\frac{(1-a)(1-b)(1-c)}{(1-d)(1-e)(1-f)}=\frac{(1-e/a)(1-f/a)}{(1-e)(1-f)}a
\\ \cross[1-\frac{(1-a)(1-d/b)(1-d/c)}{(1-d)(1-a/e)(1-a/f)}], where abc=def."
\\ Replace {a -> a/c, b -> c/b, c -> b/d, d -> f/d, e -> e/f, f -> a/e } in
\\ (8.4) to get this identity.
\\ {} 6 vars with 4 terms highest degree 2 of total 5
{ id6_4_2_5c = +b*c*(a-e)*(d-f)*(e-f) -e*f*(a-c)*(b-c)*(b-d)
+b*(c-e)*(d-f)*(a*f-c*e) +e*(a-c)*(b-f)*(b*f-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 6
{ id6_4_2_6a = +a*a*(b-f)*(c-f)*(d-f)*(e-f)
-f*f*(a-b)*(a-c)*(a-d)*(a-e)
+a*(a-f)*(d-f)*(e-f)*(a*f-b*c)
+f*(a-b)*(a-c)*(a-f)*(a*f-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ T. Amdeberhan, O. Espinosa, V. Moll, A. Straub, "Wallis-Ramanujan-Schur-
\\ Feynman", (p.5, equ. (4.3)) case n=3 : "\prod_{k=1}^n \frac{1}{y+b_k} =
\\ \sum_{k=1}^n \frac{1}{y+b_k}\prod_{j=1,j\ne k}^n\frac{1}{b_j-b_k}. "
\\ {} 6 vars with 4 terms highest degree 2 of total 6
{ id6_4_2_6b = +a*a*(c-d)*(e-f)*(c*f-d*e) -c*c*(a-b)*(e-f)*(a*f-b*e)
+e*e*(a-b)*(c-d)*(a*d-b*c) -(a*d-b*c)*(a*f-b*e)*(c*f-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ N. J. Wildberger, Divine Proportions, 2005, (p. 115, Theorem 61) : "The
\\ polynomial identity $$ (a_2b_3-a_3b_2)(a_3a_1+b_3b_1)(a_1a_2+b_1b_2) +
\\ (a_3b_1-a_1b_3)(a_2a_3+b_2b_3)(a_1a_2+b_1b_2) +(a_1b_2-a_2b_1)(a_2a_3+
\\ b_2b_3)(a_3a_1+b_3b_1) = (a_1b_2-a_2b_1)(a_2b_3-a_3b_2)(a_3b_1-a_1b_3) $$"
\\ {} 6 vars with 4 terms highest degree 2 of total 6
{ id6_4_2_6c = +(a*b+d*e)*(a*c+d*f)*(b*f-c*e) -(a*b+d*e)*(a*f-c*d)*(b*c+e*f)
+(a*c+d*f)*(a*e-b*d)*(b*c+e*f) +(a*e-b*d)*(a*f-c*d)*(b*f-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 26, equ. (8.4)): "elementary relation:
\\ 1-\frac{(1-a)(1-b)(1-c)}{(1-d)(1-e)(1-f)}=\frac{(1-e/a)(1-f/a)}{(1-e)(1-f)}a
\\ \cross[1-\frac{(1-a)(1-d/b)(1-d/c)}{(1-d)(1-a/e)(1-a/f)}], where abc=def."
\\ Replace {a -> a/c, b -> b/a, c -> c/b, d -> e/d, e -> d/f, f -> f/e } in
\\ (8.4) to get this identity.
\\ {} 6 vars with 4 terms highest degree 2 of total 6
{ id6_4_2_6d = +a*b*c*(d-e)*(d-f)*(e-f) -d*e*f*(a-b)*(a-c)*(b-c)
+f*(a-c)*(a*e-b*d)*(b*e-c*d) +b*(d-e)*(a*f-c*d)*(a*e-c*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 2 of total 6
{ id6_4_2_6e = +a*a*e*f*(b*f-c*e) -b*b*d*f*(a*f-c*d) +c*c*d*e*(a*e-b*d)
-(a*f-c*d)*(a*e-b*d)*(b*f-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 7, equ. (3.4)) case n=2 : "Let u_k, v_k
\\ and w_k be three sequences, such that $$u_k-v_k=w_k.$$ Then we have: $$
\\ \sum_k=0^n\frac{w_k}{w_0}\frac{u_0u_1\cdots u_{k-1}}{v_1v_2\cdots v_k}=
\\ \frac{u_0}{w_0}(\frac{u_1u2_\cdots u_n}{v_1v_2\cdots v_n}-\frac{v_0}{u_0})."
\\ George E. Andrews, The Theory of Partitions, 1984, (p. 106, Lemma 7.1) :
\\ "Proof. Noting that $$q^{-in}(1-x^iq^{2ni})-q^{-(i-1)n}(1-x^{i-1}q^{2n(i-1)})
\\ =q^{-in}(1-q^n) +x^{i-1}q^{n(i-1)}(1-xq^n), $$ we see that [[...]]".
\\ Replace {q^{2in}x^i -> b/e, xq^n -> d/c, q^n -> f/a}.
\\ {} 6 vars with 4 terms highest degree 3 of total 3 [NC]
{ id6_4_3_3 = +(a*b*c-d*e*f) -a*(b-e)*d -(a-f)*e*d -a*b*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 4
{ id6_4_3_4a = +a*a*(c*e-d*f) -a*(a-b)*(c*e-d*f)
+b*d*f*(a-b) -b*(a*c*e-b*d*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J. H. Silverman, The Arithmetic of Dynamical Systems, Springer, 2007.
\\ p. 190, "We also note the formal identity $$ (X-1)^2-(XY-1)(XZ-1) =
\\ X(X+Y+Z-2-XYZ) $$
\\ {} 6 vars with 4 terms highest degree 3 of total 4
{ id6_4_3_4b = +a*(a*b*c-c*d*e+d*e*f) -a*f*(a*e+b*d-d*e)
-(a*b-d*e)*(a*c-d*f) +e*f*(a-d)*(a-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 4
{ id6_4_3_4c = +a*(d*d*e-b*d*f-c*e*f+a*f*f) -c*(a*e*f+b*d*e-c*e*e-b*b*f)
+(a*d-b*c)*(b*f-d*e) -(a*f-c*e)*(a*f-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 5
{ id6_4_3_5a = +(a*b*c-d*e*f)*(d-f)*(e-f) -d*e*(a-f)*(b-f)*(c-f)
+e*f*(a-f)*(b-d)*(c-d) +f*(a-e)*(d-f)*(b*c-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 5
{ id6_4_3_5b = -(a*b*c-d*e*f)*(a*c-d*f) +b*c*f*(a-f)*(c-d)
+c*(a-f)*(a*b*c-d*e*f) +f*f*(b*c-d*e)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 6
{ id6_4_3_6a = +a*b*c*d*(b-e)*(b-f) -a*e*f*(a-b)*(b*b-c*d)
+b*b*(a-b)*(a*e*f-b*c*d) -c*d*(a*e-b*b)*(a*f-b*b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 6
{ id6_4_3_6b = +d*(a-b)*(e-f)*(a*d*e-b*c*f) -f*(a-b)*(c-d)*(a*c*f-b*d*e)
-a*a*(c-d)*(e-f)*(c*f-d*e) +(a*c-b*d)*(a*e-b*f)*(c*f-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 6
{ id6_4_3_6c = +a*a*(c-d)*(e-f)*(c*f-d*e) +c*f*(a-b)*(c-d)*(a*f-b*e)
-c*(a-b)*(a*e-b*f)*(c*f-d*e) -(a*d-b*c)*(e-f)*(a*d*e-b*c*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Albert H. Beiler, Recreations in the Theory of Numbers, Dover Publications,
\\ 1964, p. 232.
\\ {} 6 vars with 4 terms highest degree 3 of total 6
{ id6_4_3_6d = +(a*c*e+b*d*f)*(a*c*e+b*d*f) +e*f*(a*d-b*c)*(a*d-b*c)
-(a*c*e-b*d*f)*(a*c*e-b*d*f) -e*f*(a*d+b*c)*(a*d+b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Hans Reisel, Prime Numbers and Computer Methods for Factorization, 2nd ed.,
\\ BirkHauser Boston, 1994. p. 289 : "... (r_1^2-r_1s_1-Ts_1^2)(r_2^2-r_2s_2
\\ -Ts_2^2) = (r_1r_2+Ts_1s_2)^2-(r_1r_2+Ts_1s_2)(r_1s_2+s_1r_2-s_1s_2)-
\\ T(r_1s_2+s_1r_2-s_1s_2)^2 ..."
\\ {} 6 vars with 4 terms highest degree 3 of total 6
{ id6_4_3_6e = +(a*c*e-b*d*f)*(a*c*e-b*d*f)
+e*(a*c*e-b*d*f)*(a*d+b*c+b*d)
+e*f*(a*d+b*c+b*d)*(a*d+b*c+b*d)
-(a*a*e+a*b*e+b*b*f)*(c*c*e+c*d*e+d*d*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 3 of total 6
{ id6_4_3_6f = -a*a*b*b*c*c +c*d*d*(e+f)*(a*b-d*d)
+(a*b*c-d*d*e)*(a*b*c-d*d*f) +d*d*d*d*(c*e+c*f-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ George E. Andrews, The Theory of Partitions, 1984, (p. 209, Example 3) :
\\ "The proof relies on the crucial observation that $$ \frac{1}{1-xu} +\frac
\\ {1}{1-yu} -1 =\frac{1-xyu^2}{(1-xu)(1-yu)} $$ is a special case of"
\\ {} 6 vars with 4 terms highest degree 4 of total 4 [NC]
{ id6_4_4_4 = +(a*b*c*c-d*e*f*f) -a*c*(b*c-e*f) -b*c*(a*c-d*f)
+(a*c-d*f)*(b*c-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 4 of total 5
{ id6_4_4_5 = +a*a*b*b*c -b*d*f*(a*e+b*d) +(a*e+b*d)*(a*e+b*d)*f
-a*(a*b*b*c+a*e*e*f+b*d*e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 4 of total 7
{ id6_4_4_7 = +a*a*c*(b-f)*(c-f)*(d-f)*(e-f)
-f*f*c*(a-b)*(a-c)*(a-d)*(a-e)
+(a-c)*(a-f)*(c-f)*(a*a*f*f-b*c*d*e)
+(a-f)*(a*f-b*c)*(a*f-c*d)*(a*f-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 4 of total 9
{ id6_4_4_9 = +a*a*(b-f)*(c-f)*(d-f)*(e-f)*(a*a*f-b*c*d)
-f*f*(a-b)*(a-c)*(a-d)*(a-e)*(a*a*f-b*c*d)
-a*(a-f)*(e-f)*(a*f-b*c)*(a*f-b*d)*(a*f-c*d)
+f*(a-b)*(a-c)*(a-d)*(a-f)*(a*a*f*f-b*c*d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 26, equ. (8.5)).
\\ Replace {a -> a/b, b -> b/c, c -> c/d, d -> d/e, e -> e/f, f -> f/a}.
\\ {} 6 vars with 4 terms highest degree 4 of total 15
{ id6_4_4_15 =
-(a*a+a*b+b*b)*(a*a-b*f)*(a*c-b*b)*(a*d-b*c)*(a*e-b*d)*(a*f-b*e)*(a*a*e-b*b*b)
+a*a*b*b*(a+b)*(b-c)*(c-d)*(d-e)*(e-f)*(a-f)*(a*a+b*b)*(a*a*e-b*b*b)
+(a*a-b*e)*(a*c-b*b)*(a*d-b*c)*(a*e-b*d)*(a*a*f-b*b*b)*(a*a*a*e-b*b*b*f)
-a*b*(a-b)*(a+b)*(a-f)*(e-f)*(a*a+b*b)*(a*d-b*b)*(a*e-b*c)*(a*c*e-b*b*d)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From Tito Piezas III site "https://sites.google.com/site/tpiezas/010"
\\ 1. Sum of two cubes: x^3+y^3=z^3 : " ... However, in general,
\\ a (-bxy^3-cxz^3)^3 + b (ax^3y+cyz^3)^3 = c (-ax^3z+by^3z)^3,
\\ if ax^3+by^3 = cz^3 ...". Replace {c -> -c, x -> d, y -> e, z -> f}.
\\ Note: this identity is derived from id3_4_1_4c by substitution.
\\ {} 6 vars with 4 terms highest degree 4 of total 16
{ id6_4_4_16a =
+a*d*d*d*(b*e*e*e-c*f*f*f)*(b*e*e*e-c*f*f*f)*(b*e*e*e-c*f*f*f)
-b*e*e*e*(a*d*d*d-c*f*f*f)*(a*d*d*d-c*f*f*f)*(a*d*d*d-c*f*f*f)
+c*f*f*f*(a*d*d*d-b*e*e*e)*(a*d*d*d-b*e*e*e)*(a*d*d*d-b*e*e*e)
+(a*d*d*d+b*e*e*e+c*f*f*f)*(a*d*d*d-b*e*e*e)*(a*d*d*d-c*f*f*f)*(b*e*e*e-c*f*f*f)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 589) :
\\ "A. M. Legendre$^{288}$ deduced from one solution of x^3+ay^3=bz^3 the
\\ second solution X = x(x^3+2ay^3), Y = -y(2x^3+ay^3), Z = z(x^3-ay^3)."
\\ Note: this identity is derived from id3_4_1_4i by substitution.
\\ {} 6 vars with 4 terms highest degree 4 of total 16
{ id6_4_4_16b =
+a*d*d*d*(b*e*e*e+b*e*e*e+a*d*d*d)*(b*e*e*e+b*e*e*e+a*d*d*d)*(b*e*e*e+b*e*e*e+a*d*d*d)
-b*e*e*e*(a*d*d*d+a*d*d*d+b*e*e*e)*(a*d*d*d+a*d*d*d+b*e*e*e)*(a*d*d*d+a*d*d*d+b*e*e*e)
+c*f*f*f*(a*d*d*d-b*e*e*e)*(a*d*d*d-b*e*e*e)*(a*d*d*d-b*e*e*e)
-(a*d*d*d+b*e*e*e+c*f*f*f)*(a*d*d*d-b*e*e*e)*(a*d*d*d-b*e*e*e)*(a*d*d*d-b*e*e*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 5 of total 15
{ id6_4_5_15a =
+a*a*(b-f)*(b-f)*(c-f)*(d-f)*(e-f)*(a*a*f-b*b*c)*(a*a*a*f*f-b*b*c*d*e)
+f*f*(a-b)*(a-b)*(a-c)*(a-d)*(a-e)*(a*a*f-b*b*c)*(a*a*f*f*f-b*b*c*d*e)
-a*(a-f)*(d-f)*(e-f)*(a*f-b*b)*(a*f-b*c)*(a*f-b*c)*(a*a*a*f*f-b*b*c*d*e)
-f*(a-b)*(a-b)*(a-c)*(a-f)*(a*f-d*e)*(a*a*f*f-b*b*c*d)*(a*a*f*f-b*b*c*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 5 of total 15
{ id6_4_5_15b =
+a*a*(b-f)*(b-f)*(c-f)*(d-f)*(e-f)*(a*a*f-b*c*d)*(a*a*a*f*f-b*b*c*d*e)
+f*f*(a-b)*(a-b)*(a-c)*(a-d)*(a-e)*(a*a*f-b*c*d)*(a*a*f*f*f-b*b*c*d*e)
-a*(a-f)*(b-f)*(e-f)*(a*f-b*c)*(a*f-b*d)*(a*f-c*d)*(a*a*a*f*f-b*b*c*d*e)
-f*(a-b)*(a-c)*(a-d)*(a-f)*(a*f-b*e)*(a*a*f*f-b*b*c*d)*(a*a*f*f-b*c*d*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 6 of total 19
{ id6_4_6_19a =
+a*a*b*(b-f)*(b-f)*(c-f)*(d-f)*(e-f)*(a*a*a*f*f-b*b*c*d*e)*(a*a*a*f*f*f-b*b*b*c*d*e)
+b*f*f*(a-b)*(a-b)*(a-c)*(a-d)*(a-e)*(a*a*f*f*f-b*b*c*d*e)*(a*a*a*f*f*f-b*b*b*c*d*e)
-(a-b)*(a-f)*(b-f)*(a*a*f*f-b*b*c*d)*(a*a*f*f-b*b*c*e)*(a*a*f*f-b*b*d*e)*(a*a*f*f-b*c*d*e)
-(a-f)*(a*f-b*b)*(a*f-b*c)*(a*f-b*d)*(a*f-b*e)*(a*a*a*f*f-b*b*c*d*e)*(a*a*f*f*f-b*b*c*d*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 6 of total 19
{ id6_4_6_19b =
+a*a*c*(b-f)*(b-f)*(c-f)*(d-f)*(e-f)*(a*a*a*f*f-b*b*c*d*e)*(a*a*a*f*f*f-b*b*c*c*d*e)
+c*f*f*(a-b)*(a-b)*(a-c)*(a-d)*(a-e)*(a*a*f*f*f-b*b*c*d*e)*(a*a*a*f*f*f-b*b*c*c*d*e)
-(a-c)*(a-f)*(c-f)*(a*a*f*f-b*b*c*d)*(a*a*f*f-b*b*c*e)*(a*a*f*f-b*c*d*e)*(a*a*f*f-b*c*d*e)
-(a-f)*(a*f-b*c)*(a*f-b*c)*(a*f-c*d)*(a*f-c*e)*(a*a*a*f*f-b*b*c*d*e)*(a*a*f*f*f-b*b*c*d*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 4 terms highest degree 6 of total 19
{ id6_4_6_19c =
+a*a*d*(b-f)*(b-f)*(c-f)*(d-f)*(e-f)*(a*a*a*f*f-b*b*c*d*e)*(a*a*a*f*f*f-b*b*c*d*d*e)
+d*f*f*(a-b)*(a-b)*(a-c)*(a-d)*(a-e)*(a*a*f*f*f-b*b*c*d*e)*(a*a*a*f*f*f-b*b*c*d*d*e)
-(a-d)*(a-f)*(d-f)*(a*a*f*f-b*b*c*d)*(a*a*f*f-b*b*d*e)*(a*a*f*f-b*c*d*e)*(a*a*f*f-b*c*d*e)
-(a-f)*(a*f-b*d)*(a*f-b*d)*(a*f-c*d)*(a*f-d*e)*(a*a*a*f*f-b*b*c*d*e)*(a*a*f*f*f-b*b*c*d*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 1 of total 2 [NC]
{ id6_5_1_2a = +(a-c)*(e-f) +(a-f)*(b-e) -(a-f)*(b-f)
-(c-f)*(d-e) +(d-f)*(c-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 1 of total 2
{ id6_5_1_2b = -(a-c)*(b-d) +(a-f)*(b-c) +(a-f)*(c-d)
-(b-e)*(c-f) +(c-f)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 1 of total 2
{ id6_5_1_2c = +(a-e)*(b-e) -(a-f)*(b-f) +(c-f)*(d-f)
-(c-e)*(d-e) +(a+b-c-d)*(e-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id6_4_3_3 with its first term split into two terms.
\\ B. C. Berndt, Ramanujan's Notebooks, Part IV, (p. 37, (26.1)) case r=3 :
\\ "$$ \sum_{j=0}^{r-1} (a_j-b_{j+1})\left(\frac{b_1\dots b_j}{a_1\dots a_j}
\\ \right) = a_0 -\left(\frac{b_1\dots\b_r}{a_1\dots a_{r-1}}\right). $$
\\ J. M. Steele, The Cauchy-Schwarz Master Class, 2004 (p. 276) case n=3 :
\\ "Solution for Exercise 12.8. [[...]] Here the telescoping identity $$
\\ a_1a_2\cdots a_n-b_1b_2\cdots b_n = \sum_{j=1}^n a_1\cdots a_{j-1}(a_j-b_j)
\\ b_{j+1}\cdots b_n $$ makes the Weierstrass inequality immediate."
\\ J. W. L. Glaisher, On the deduction of series from infinite products,
\\ Messenger of Math., 2 (1873), p. 138. "to make use of the identities
\\ (1-a)(1-b)(1-c)... = 1-a-b(1-a)-c(1-a)(1-b)-... (2)"
\\ {} 6 vars with 5 terms highest degree 1 of total 3 [NC]
{ id6_5_1_3a = +a*b*c -d*e*f -(a-d)*e*f -a*(b-e)*f -a*b*(c-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 1 of total 3
{ id6_5_1_3b = +(a-b)*(a-c)*(e-f) -(a-b)*(a-e)*(b-f) +(a-b)*(c-e)*(d-f)
+(a-c)*(a-d)*(b-e) -(a-e)*(b-c)*(b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 1 of total 3
{ id6_5_1_3c = +a*(b-f)*(d-e) +a*(c-d)*(e-f) -a*(c-e)*(d-f)
-b*(a-c)*(d-e) +c*(a-b)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 1 of total 3
{ id6_5_1_3d = +(a-d)*(b-f)*(c-f) -(a-f)*(b-f)*(c-f) +(b-f)*(d-f)*(e-f)
+(b-c)*(c-e)*(d-f) +(c-e)*(c-f)*(d-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gasper+Rahman, Basic Hypergeometric Series 2nd ed. (p. 331, equ. (11.7.3))
\\ case n=3 : "[[...]] is the easily verifiable identity
\\ \sum_{k=1}^n\frac{\prod_{j=1}^n(b_j-a_k)}{a_k\prod_{j\ne k}(a_j-a_k)} =
\\ \frac{b_1...b_n}{a_1...a_n}-1. "
\\ {} 6 vars with 5 terms highest degree 1 of total 6
{ id6_5_1_6 = +a*b*c*(a-b)*(a-c)*(b-c) -d*e*f*(a-b)*(a-c)*(b-c)
-a*b*(a-b)*(c-d)*(c-e)*(c-f) +a*c*(a-c)*(b-d)*(b-e)*(b-f)
-b*c*(a-d)*(a-e)*(a-f)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Max Alekseyev, Math Overflow question 280195, Sep 02 2017 case n=3:
\\ $$ (k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n
\\ \frac{t-d_j}{d_i-d_j}, $$
\\ {} 6 vars with 5 terms highest degree 1 of total 9 [TS]
{ id6_5_1_9 = +(a-f)*(a-f)*(a-f)*(b-c)*(b-d)*(b-e)*(c-d)*(c-e)*(d-e)
-(b-f)*(b-f)*(b-f)*(a-c)*(a-d)*(a-e)*(c-d)*(c-e)*(d-e)
+(c-f)*(c-f)*(c-f)*(a-b)*(a-d)*(a-e)*(b-d)*(b-e)*(d-e)
-(d-f)*(d-f)*(d-f)*(a-b)*(a-c)*(a-e)*(b-c)*(b-e)*(c-e)
+(e-f)*(e-f)*(e-f)*(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is part of Strassen multiplication of 2x2 matrices.
\\ {} 6 vars with 5 terms highest degree 2 of total 2 [NC]
{ id6_5_2_2a = +(a*b-c*d) -(a-e)*(b-f) +(c-e)*(d-f) -e*(b-d) -(a-c)*f ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into five rectangles.
\\ This is the Summation by Parts case n=3 available at
\\ "http://dlmf.nist.gov/2.10#E9" and other sources.
\\ {} 6 vars with 5 terms highest degree 2 of total 2 [NC]
{ id6_5_2_2b = -(a*b-e*f) +a*(b-d) +c*(d-f) +(a-c)*d +(c-e)*f ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 2 of total 2 [NC]
{ id6_5_2_2c = -(d+e+f)*(a+b+c) +d*(a+b) +e*(b+c) +f*(a+c) +(e*a+f*b+d*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 2 of total 3 [NC]
{ id6_5_2_3a = -(d+e+f)*(a*a+b*b+c*c+a*b+a*c+b*c) +d*(a*a+a*b+b*b)
+e*(b*b+b*c+c*c) +f*(a*a+a*c+c*c) +(e*a+b*f+d*c)*(a+b+c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Eduardo Casas-Alvero, Analytic Projective Geometry, 2014, (p. 25) :
\\ "Since $$ \lambda\mu(u+v+w)-(u+\lambda v+\mu w)-(\lambda\mu-1)u-\lambda
\\ (\mu-1)v-\mu(\lambda-1)w = 0, $$ the claim is proved."
\\ Replace {\lambda -> a/f, \mu -> b/f, u -> c, v -> d, w -> e}.
\\ {} 6 vars with 5 terms highest degree 2 of total 3
{ id6_5_2_3b = +a*d*(b-f) +b*e*(a-f) +c*(a*b-f*f) +f*(a*d+b*e+c*f)
-a*b*(c+d+e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Lagrange's identity for n=3. Used to prove the Cauchy-Schwarz inequality.
\\ For example in https://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality.
\\ Also refer to https://en.wikipedia.org/wiki/Lagrange's_identity.
\\ {} 6 vars with 5 terms highest degree 2 of total 4
{ id6_5_2_4a = +(a*a+b*b+c*c)*(d*d+e*e+f*f) -(a*d+b*e+c*f)*(a*d+b*e+c*f)
-(a*e-b*d)*(a*e-b*d) -(b*f-c*e)*(b*f-c*e) -(c*d-a*f)*(c*d-a*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {a -> i*a, d -> i*d} where i^2 = -1 to get id6_5_2_4a.
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 16) case n=3 :
\\ "Aczel's Inequality. If $a,b$ are real n-tuples with $a_1^2-\sum_{i=2}^n
\\ a_i^2>0$ then $$ (a_1^2-\sum_{i=2}^n a_i^2)(b_1^2-\sum_{i=2}^n b_i^2)\le
\\ (a_1b_1-\sum_{i=2}^n a_ib_i)^2 $$, with equality if and only if $a~b$."
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 263) :
\\ "T. Weddle^{26} noted that , if $(a,p,z),(b,q,z')$ and $(c,r,z'')$ are
\\ the extremities of a system of conjugate semiaxes of an ellipsoid,
\\ $$ (a^2+b^2+c^2)(p^2+q^2+r^2) = (aq-bp)^2 + (ar-cp)^2 + (br-cq)^2. $$
\\ {} 6 vars with 5 terms highest degree 2 of total 4
{ id6_5_2_4b = +(a*a-b*b-c*c)*(d*d-e*e-f*f) -(a*d-b*e-c*f)*(a*d-b*e-c*f)
+(a*e-b*d)*(a*e-b*d) -(b*f-c*e)*(b*f-c*e) +(c*d-a*f)*(c*d-a*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 2 of total 6
{ id6_5_2_6 = +a*b*c*f*(d-f)*(e-f) -a*d*e*f*(b-f)*(c-f)
+d*e*(a*b-f*f)*(a*c-f*f) -b*c*(a*d-f*f)*(a*e-f*f)
-f*f*f*(a-f)*(b*c-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 2 of total 7
{ id6_5_2_7a = +a*b*c*f*(c-f)*(d-f)*(e-f) -c*d*e*f*(a-f)*(a-f)*(b-f)
+d*e*(a-f)*(a*c-f*f)*(b*c-f*f) -b*c*(c-f)*(a*d-f*f)*(a*e-f*f)
-f*f*f*(a-f)*(c-f)*(b*c-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 2 of total 7
{ id6_5_2_7b = +a*b*d*f*(c-f)*(d-f)*(e-f) -a*d*e*f*(a-f)*(b-f)*(c-f)
+a*e*(a-f)*(b*d-f*f)*(c*d-f*f) -b*d*(d-f)*(a*c-f*f)*(a*e-f*f)
+f*f*f*(a-f)*(d-f)*(a*e-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 2 of total 7
{ id6_5_2_7c = +a*c*d*f*(b-f)*(d-f)*(e-f) -a*d*e*f*(a-f)*(b-f)*(c-f)
+a*e*(a-f)*(b*d-f*f)*(c*d-f*f) -c*d*(d-f)*(a*b-f*f)*(a*e-f*f)
+f*f*f*(a-f)*(d-f)*(a*e-c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 4 [NC]
{ id6_5_3_4 =
-(d+e+f)*(a*a*a+a*a*b+a*b*b+b*b*b+a*a*c+a*c*c+c*c*c+b*b*c+b*c*c+a*b*c)
+d*(a*a*a+a*a*b+a*b*b+b*b*b)
+e*(b*b*b+b*b*c+b*c*c+c*c*c)
+f*(a*a*a+a*a*c+a*c*c+c*c*c)
+(e*a+b*f+d*c)*(a*a+b*b+c*c+a*b+a*c+b*c)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 5
{ id6_5_3_5 = +a*(c*e+d*f)*(c*e+d*f) -b*(c*e+d*f)*(a*e+b*f)
-c*(a*e+b*f)*(c*e+d*f) +d*(a*e+b*f)*(a*e+b*f)
-(a*d-b*c)*(a*e*e+b*e*f+c*e*f+d*f*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 6
{ id6_5_3_6a = +(a-d)*(b-e)*(c-f)*(a*b*c+d*e*f) -(a*b-d*e)*(a*c-d*f)*(b*c-e*f)
+a*d*(b-e)*(c-f)*(b*c-e*f) +b*e*(a-d)*(c-f)*(a*c-d*f)
+c*f*(a-d)*(b-e)*(a*b-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 6
{ id6_5_3_6b = +a*a*b*b*c*c -d*d*e*e*f*f -a*b*c*(a*e*f+b*d*f+c*d*e)
+d*e*f*(a*b*f+a*c*e+b*c*d) -(a*b-d*e)*(a*c-d*f)*(b*c-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 7
{ id6_5_3_7a = +a*b*c*d*f*(d-f)*(e-f) -a*a*d*e*f*(b-f)*(c-f)
+a*a*e*(b*d-f*f)*(c*d-f*f) -b*c*d*(a+f)*(d-f)*(a*e-f*f)
+f*f*f*(d-f)*(a*a*e-b*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 7
{ id6_5_3_7b = +a*b*b*c*f*(d-f)*(e-f) -a*b*d*e*f*(a-f)*(c-f)
+a*d*e*(a-f)*(b+f)*(b*c-f*f) -b*b*c*(a*d-f*f)*(a*e-f*f)
+f*f*f*(a-f)*(a*d*e-b*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 7
{ id6_5_3_7c = +a*b*c*c*f*(d-f)*(e-f) -a*c*d*e*f*(a-f)*(b-f)
+a*d*e*(a-f)*(c+f)*(b*c-f*f) -b*c*c*(a*d-f*f)*(a*e-f*f)
+f*f*f*(a-f)*(a*d*e-b*c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 8
{ id6_5_3_8 = +a*b*b*c*f*(c-f)*(d-f)*(e-f) -a*c*d*e*f*(a-f)*(b-f)*(b-f)
+a*d*e*(a-f)*(b*c-f*f)*(b*c-f*f) -b*b*c*(c-f)*(a*d-f*f)*(a*e-f*f)
+f*f*f*(a-f)*(c-f)*(a*d*e-b*b*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 3 of total 9
{ id6_5_3_9 = -(a*b*c-d*e*f)*(a*c*e-b*d*f)*(a*c-d*f)*e
+c*c*e*(a-f)*(a*e-b*f)*(a*b*c-d*e*f)
+b*c*c*f*(a-f)*(b+e)*(c-d)*(a*e-b*f)
+e*f*f*(c-d)*(b*c-d*e)*(a*c*e-b*d*f)
+b*c*f*f*(a-f)*(b+e)*(c-d)*(b*c-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Allan Cunningham, _Binomial Factorisations_, Vol. 1, p. 427. Replace
\\ {a' -> c, b' -> d, x -> e, y ->f} to get this identity.
\\ {} 6 vars with 5 terms highest degree 4 of total 6
{ id6_5_4_6 = +(a*d+b*c)*e*e*e*f +(a*a+c*c+b*d)*e*e*f*f
+(a*b+c*d)*e*f*f*f +a*c*(e*e*e*e+f*f*f*f)
-(a*e*e+b*e*f+c*f*f)*(c*e*e+d*e*f+a*f*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 5 terms highest degree 4 of total 8
{ id6_5_4_8 = +d*d*f*(b*c-e*f)*(b*c*c-e*f*f)
+b*b*c*c*f*(a-d)*(a*c-d*f)
+c*d*f*(a-d)*(b+e)*(c+f)*(b*c-e*f)
-d*e*(c+f)*(a*c-d*f)*(b*c*c-e*f*f)
-c*(a*b*f-c*d*e)*(a*b*c*c-d*e*f*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id6_5_2_2a with its first term split into two terms.
\\ {} 6 vars with 6 terms highest degree 1 of total 2 [NC]
{ id6_6_1_2a = +a*b -c*d -(a-e)*(b-f) +(c-e)*(d-f) -e*(b-d) -(a-c)*f ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is id6_5_2_2b with its first term split into two terms.
\\ This is a dissection of a rectangle into five rectangles.
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 13) case n=3 :
\\ "Abel's Summation Formula, $$ \sum_{i=1}^n w_ia_i = W_n a_n - \sum_{i=1}
\\ ^{n-1} W_i\Delta a_i. (2) $$".
\\ Replace {w_1 -> a, w_2 -> e-a, w_3 -> -c, a_1 -> b, a_2 ->f, a_3 -> d}.
\\ B. C. Berndt, Ramanujan's Notebooks, Part IV, (p. 36., (25.1)) case n=1 :
\\ {} 6 vars with 6 terms highest degree 1 of total 2 [NC]
{ id6_6_1_2b = +a*b -c*d -a*(b-f) +e*(d-f) -(a-e)*f +(c-e)*d ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 6 terms highest degree 1 of total 2 [NC]
{ id6_6_1_2c = +(a-b)*(d+e) +(b-c)*(e+f) -(a-c)*(d+f)
-a*(e-f) +b*(d-f) -c*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the determinant of the 3x3 rank 2 matrix
\\ [a-b, a-d, a-f; c-b, c-d, c-f; e-b, e-d, e-f]
\\ Given a 3x3 magic square [ a, b, c; d, e, f; g, h, i] then we have
\\ a*b*c + d*e*f + g*h*i = a*d*g + b*e*h + c*f*i. This identity comes from
\\ the determinant of [ a, f, h; i, b, d; e, g, c]. See pages 64 to 65 of
\\ Edward J. Barbeau, Power Play, MAA, 1997.
\\ {} 6 vars with 6 terms highest degree 1 of total 3 [TS]
{ id6_6_1_3a = +(a-b)*(c-d)*(e-f) +(a-b)*(d-e)*(c-f) -(a-d)*(b-e)*(c-f)
+(a-d)*(b-c)*(e-f) +(a-f)*(b-c)*(d-e) +(a-f)*(b-e)*(c-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 6 terms highest degree 1 of total 3
{ id6_6_1_3b = +a*b*(c-f) +a*c*(b-e) +e*f*(a-d) -a*(b-e)*(c-f) -a*b*c +d*e*f ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the determinant of the 3x3 rank 2 matrix
\\ [a-b+d-e, b-c+d-e, a-c-d+e; a-b+e-f, b-c+e-f, a-c-e+f; a-b-d+f, b-c-d+f, a-c+d-f]
\\ {} 6 vars with 6 terms highest degree 1 of total 3 [JE]
{ id6_6_1_3c = +(a-b-d+f)*(a-c-d+e)*(b-c+e-f) -(a-b-d+f)*(a-c-e+f)*(b-c+d-e)
-(a-b+e-f)*(a-c-d+e)*(b-c-d+f) +(a-b+d-e)*(a-c-e+f)*(b-c-d+f)
+(a-b+e-f)*(a-c+d-f)*(b-c+d-e) -(a-b+d-e)*(a-c+d-f)*(b-c+e-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the determinant of the 3x3 rank 2 matrix
\\ [a-b+d-e, b-c+d-e, c-e; a-b+e-f, b-c+e-f, c-d+e-f; b-f, a-b+c-f, a-c+d-f]
\\ {} 6 vars with 6 terms highest degree 1 of total 3 [JE]
{ id6_6_1_3d = +(a-b+c-f)*(a-b+e-f)*(c-e) +(a-b+d-e)*(a-c+d-f)*(b-c+e-f)
-(a-b+e-f)*(a-c+d-f)*(b-c+d-e) -(a-b+c-f)*(a-b+d-e)*(c-d+e-f)
+(b-c+d-e)*(b-f)*(c-d+e-f) -(b-c+e-f)*(b-f)*(c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ B. C. Berndt, Ramanujan's Notebooks, Part IV, (p. 36., (25.1)) case n=4 :
\\ "$$ \sum_{k=0}^{n-1} \frac{a_0a_1\dots a_k}{(x+a_1)(x+a_2)\dots(x+a_{k+1})}
\\ = \frac{1}{x}-\frac{1}{x(1+x/a_1)(1+x/a_2)\dots(1+x/a_n)}. $$"
\\ {} 6 vars with 6 terms highest degree 1 of total 4 [NC]
{ id6_6_1_4 = +(a-b)*(a-c)*(a-d)*(a-e) -(a-f)*(a-c)*(a-d)*(a-e)
+(b-f)*(a-f)*(a-d)*(a-e) -(b-f)*(c-f)*(a-f)*(a-e)
+(b-f)*(c-f)*(d-f)*(a-f) -(b-f)*(c-f)*(d-f)*(e-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the determinant of a 3x3 matrix similar to id6_6_1_3a.
\\ R. Wm. Gosper and R. Schroeppel, "Somos Sequence Near-Addition Formulas
\\ and Modular Theta Functions" arXiv:math/0703470v1 (p. 4, Conjecture 4)
\\ {} 6 vars with 6 terms highest degree 1 of total 6 [WS]
{ id6_6_1_6 = +(a-d)*(a+d)*(b-e)*(b+e)*(c-f)*(c+f)
+(a-e)*(a+e)*(b-f)*(b+f)*(c-d)*(c+d)
+(a-f)*(a+f)*(b-d)*(b+d)*(c-e)*(c+e)
-(a-d)*(a+d)*(b-f)*(b+f)*(c-e)*(c+e)
-(a-e)*(a+e)*(b-d)*(b+d)*(c-f)*(c+f)
-(a-f)*(a+f)*(b-e)*(b+e)*(c-d)*(c+d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ T. Amdeberhan, O. Espinosa, V. Moll, A. Straub, "Wallis-Ramanujan-Schur-
\\ Feynman", (p.5, equ. (4.3)) case n=5 : "\prod_{k=1}^n \frac{1}{y+b_k} =
\\ \sum_{k=1}^n \frac{1}{y+b_k}\prod_{j=1,j\ne k}^n\frac{1}{b_j-b_k}. "
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=6, r=0 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 6 vars with 6 terms highest degree 1 of total 10
{ id6_6_1_10 = +(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e)
-(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-f)*(b-f)*(c-f)*(d-f)
+(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)*(a-f)*(b-f)*(c-f)*(e-f)
-(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)*(a-f)*(b-f)*(d-f)*(e-f)
+(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)*(a-f)*(c-f)*(d-f)*(e-f)
-(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)*(b-f)*(c-f)*(d-f)*(e-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=6, r=1 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 6 vars with 6 terms highest degree 1 of total 11
{ id6_6_1_11 =
-a*(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)*(b-f)*(c-f)*(d-f)*(e-f)
+b*(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)*(a-f)*(c-f)*(d-f)*(e-f)
-c*(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)*(a-f)*(b-f)*(d-f)*(e-f)
+d*(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)*(a-f)*(b-f)*(c-f)*(e-f)
-e*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-f)*(b-f)*(c-f)*(d-f)
+f*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=6, r=2 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 6 vars with 6 terms highest degree 1 of total 12
{ id6_6_1_12 =
-a*a*(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)*(b-f)*(c-f)*(d-f)*(e-f)
+b*b*(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)*(a-f)*(c-f)*(d-f)*(e-f)
-c*c*(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)*(a-f)*(b-f)*(d-f)*(e-f)
+d*d*(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)*(a-f)*(b-f)*(c-f)*(e-f)
-e*e*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-f)*(b-f)*(c-f)*(d-f)
+f*f*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=6, r=3 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 6 vars with 6 terms highest degree 1 of total 13
{ id6_6_1_13 =
-a*a*a*(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)*(b-f)*(c-f)*(d-f)*(e-f)
+b*b*b*(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)*(a-f)*(c-f)*(d-f)*(e-f)
-c*c*c*(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)*(a-f)*(b-f)*(d-f)*(e-f)
+d*d*d*(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)*(a-f)*(b-f)*(c-f)*(e-f)
-e*e*e*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-f)*(b-f)*(c-f)*(d-f)
+f*f*f*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=6, r=4 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 6 vars with 6 terms highest degree 1 of total 14 [TS]
{ id6_6_1_14 =
-a*a*a*a*(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)*(b-f)*(c-f)*(d-f)*(e-f)
+b*b*b*b*(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)*(a-f)*(c-f)*(d-f)*(e-f)
-c*c*c*c*(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)*(a-f)*(b-f)*(d-f)*(e-f)
+d*d*d*d*(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)*(a-f)*(b-f)*(c-f)*(e-f)
-e*e*e*e*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-f)*(b-f)*(c-f)*(d-f)
+f*f*f*f*(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Let x = p+1/p, y = q+1/q, z = r+1/r. Then x^2 + y^2 + z^2 - x*y*z - 4 =
\\ (1 - p*q/r)*(1 - p*r/q)*(1 - q*r/p)*(1 - 1/p/q/r).
\\ Replace {p -> b/a, q -> d/c, r -> f/e} to get this identity.
\\ {} 6 vars with 6 terms highest degree 3 of total 12
{ id6_6_3_12 =
-(a+a)*(a+a)*b*b*c*c*d*d*e*e*f*f
+(a*a+b*b)*(a*a+b*b)*c*c*d*d*e*e*f*f
+a*a*b*b*(c*c+d*d)*(c*c+d*d)*e*e*f*f
+a*a*b*b*c*c*d*d*(e*e+f*f)*(e*e+f*f)
-a*b*(a*a+b*b)*c*d*(c*c+d*d)*e*f*(e*e+f*f)
-(a*c*e-b*d*f)*(a*c*f-b*d*e)*(a*d*e-b*c*f)*(a*d*f-b*c*e)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Harold M. Edwards, A Normal Form for Elliptic Curves, B.A.M.S, v. 44 n. 3,
\\ p. 405 : "The calculation will make use of the formula $$ (9.1) \frac
\\ {(1-x_1^2x^2)(1-y_1^2x^2)}{1-P^2} = [[...]] = 1 - a^2x^2 $$"
\\ {} 6 vars with 6 terms highest degree 4 of total 6
{ id6_6_4_6 = +a*a*a*a*c*(e+f) -a*a*c*e*f*(c+d) -a*a*b*c*(a*a+e*f)
+a*a*(a*a-c*e)*(a*a-c*f) -(a*a-b*c)*(a*a*a*a-c*d*e*f)
+b*c*e*f*(a*a+c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is every other term of id12_14_1_2.
\\ {} 6 vars with 7 terms highest degree 1 of total 2 [NC]
{ id6_7_1_2 = +a*d +b*e +c*f -(a+b)*(d+e) -(a+c)*(d+f) -(b+c)*(e+f)
+(a+b+c)*(d+e+f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the Cauchy Determinant case n=3 available at
\\ "http://dlmf.nist.gov/1.3#E14" and other sources.
\\ {} 6 vars with 7 terms highest degree 1 of total 6 [TS]
{ id6_7_1_6 = +(a-b)*(a-c)*(b-c)*(d-e)*(d-f)*(e-f)
-(a-d)*(a-e)*(b-d)*(b-f)*(c-e)*(c-f)
+(a-d)*(a-f)*(b-d)*(b-e)*(c-e)*(c-f)
+(a-d)*(a-e)*(b-e)*(b-f)*(c-d)*(c-f)
-(a-d)*(a-f)*(b-e)*(b-f)*(c-d)*(c-e)
-(a-e)*(a-f)*(b-d)*(b-e)*(c-d)*(c-f)
+(a-e)*(a-f)*(b-d)*(b-f)*(c-d)*(c-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 7 terms highest degree 2 of total 4
{ id6_7_2_4 = +(a*d+b*f+c*e)*(a*d+b*f+c*e) -(a*f+b*e+c*d)*(a*e+b*d+c*f)
+(a*e+b*d+c*f)*(a*e+b*d+c*f) -(a*d+b*f+c*e)*(a*e+b*d+c*f)
+(a*f+b*e+c*d)*(a*f+b*e+c*d) -(a*f+b*e+c*d)*(a*d+b*f+c*e)
-(a*a+b*b+c*c-a*b-a*c-b*c)*(d*d+e*e+f*f-d*e-d*f-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 6 vars with 8 terms highest degree 1 of total 4
{ id6_8_1_4 = +(a-f)*(b-f)*(a-b)*(c-f) -(a-f)*(d-f)*(a-d)*(c-f)
-(a-b)*(a-d)*(b-d)*(c-f) +(b-f)*(b-f)*(b-d)*(c-f)
-(b-f)*(e-f)*(b-e)*(b-d) -(b-c)*(b-e)*(c-e)*(b-d)
+(c-f)*(e-f)*(c-e)*(b-d) -(b-f)*(c-f)*(c-d)*(b-d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 7 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 3 terms highest degree 2 of total 3
{ id7_3_2_3 = -(d-e)*(a*g-b*f+c*f-c*g)
-(c-d)*(a*g-b*f-e*g+e*f)
+(c-e)*(a*g-b*f-d*g+d*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 3 terms highest degree 2 of total 8
{ id7_3_2_8 = +a*(b-c)*(a*a-b*c)*(d*g-e*f)*(d*f-e*g)
+(a*d-b*e)*(a*e-b*d)*(a*f-c*g)*(a*g-c*f)
-(a*d-c*e)*(a*e-c*d)*(a*f-b*g)*(a*g-b*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 3 terms highest degree 3 of total 5
{ id7_3_3_5 = +(a*a*g-e*f*f)*(b*d-c*c)
-(a*b*g-c*e*f)*(a*d-c*f)
+(a*c*g-d*e*f)*(a*c-b*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 3 terms highest degree 3 of total 6
{ id7_3_3_6 = +(a*e*g-b*f*g)*(c*c*e-d*d*f)
+(a*c*c-f*f*g)*(b*d*d-e*e*g)
-(a*d*d-e*f*g)*(b*c*c-e*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 3 terms highest degree 4 of total 5
{ id7_3_4_5 = +a*(b*b*d*g+b*c*c*g+b*d*d*e+c*c*d*e+c*d*d*f)
-d*(a*a*c*g+a*b*b*g+a*c*c*e+b*b*c*e+b*c*c*f)
+(a*d-b*c)*(a*c*g-b*d*e-c*d*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Clapperton, Larcombe and Fennessey, "Two New Identities for Polynomial
\\ Families", Bulletin of the ICA, 62 (2011), pp. 25-32. (p. 29 equ. (II.2))
\\ case n=2 : "It can be shown inductively that an arbitrary recurrence
\\ $$ y_n = \delta_nz_{n-1}+\epsilon_nz_n, n\ge 1. (II.1) $$ leads to the
\\ following identity: $$ \sum_{i=1}^n \left\[\prod_{j=i+1}^n\delta_j
\\ \prod_{l=1}^{i-1}\epsilon_l\right\](-1)^{i+1}y_i = z_0\prod{j=1}^n\delta_j
\\ +(-1)^{n+1}z_n\prod_{l=1}^n\epsilon_l. (II.2) $$"
\\ {} 7 vars with 4 terms highest degree 2 of total 3
{ id7_4_2_3a = -a*b*c +d*e*f +a*(b*c-d*g) +d*(a*g-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 2 of total 3 [NC]
{ id7_4_2_3b = +a*(g*b-e*c) -(a*g+c*d)*b +(a*e+c*f)*c +c*(d*b-f*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ "A very elementary proof that the Somos 5 sequence is integer valued" at
\\ "http://www.maths.ed.ac.uk/~mwemyss/Somos5proof.pdf" (p. 1 Lemma 2.4.)
\\ " [...] a_{n-3}(a_n^2+a_{n-2}a{n+2}) = [...] a_{n+1}(a_{n-2}^2+a_na_{n-4})
\\ where $a_na_{n+5} = a_{n+1}a_{n+4} + a_{n+2}a_{n+3} for n = 0,1,2,...$.
\\ {} 7 vars with 4 terms highest degree 2 of total 3
{ id7_4_2_3c = +b*(c*g+e*e) -c*(b*g-c*f-d*e) +e*(a*f-b*e-c*d) -f*(a*e+c*c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 2 of total 4
{ id7_4_2_4 = +c*d*(a-b)*(e-f) -d*(a-b)*(c*e-f*g)
+f*(d-g)*(a*c-b*d) +f*(c-d)*(a*g-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 2 of total 5
{ id7_4_2_5a = +a*(b-c)*(d-e)*(a*g-e*f) -e*(a-d)*(f-g)*(a*b-c*e)
+c*g*(a-d)*(a-e)*(d-e) -(a-e)*(a*b-c*d)*(d*g-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 2 of total 5
{ id7_4_2_5b = +d*(a-d)*(f-g)*(a*b-c*e) -d*(b-c)*(d-e)*(a*g-e*f)
-b*f*(a-d)*(a-e)*(d-e) +(a-e)*(a*b-c*d)*(d*g-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 2 of total 5
{ id7_4_2_5c = +a*(a-b)*(f-g)*(b*c-d*e) +a*(a-e)*(c-d)*(b*g-e*f)
-c*f*(a-b)*(a-e)*(b-e) +(b-e)*(a*d-b*c)*(a*g-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 2 of total 5
{ id7_4_2_5d = +d*g*(a-b)*(a-e)*(b-e) -b*(a-e)*(c-d)*(b*g-e*f)
-e*(a-b)*(f-g)*(b*c-d*e) -(b-e)*(a*d-b*c)*(a*g-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 3 of total 4
{ id7_4_3_4a = +a*d*(b-f)*(c-g) +b*c*(a-d)*(a-e) +(a-d)*(b*c*e-d*f*g)
-(a*b-d*f)*(a*c-d*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 3 of total 4
{ id7_4_3_4b = +a*d*(b-f)*(c-g) -b*c*(a-e)*(d-f) -a*(b-d)*(c*f-d*g)
+(d-f)*(a*d*g-b*c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 3 of total 5
{ id7_4_3_5a = +a*b*e*(c-f)*(d-g) +a*c*d*(a-e)*(b-f) -b*(a*c-e*f)*(a*d-e*g)
+f*(a-e)*(a*c*d-b*e*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 3 of total 5
{ id7_4_3_5b = +a*(a*b-c*f)*(b*g-d*e) -g*(a*b-c*f)*(a*b-c*f)
+(b*d-c*c)*(a*a*e-f*f*g) +(a*c-d*f)*(a*c*e-b*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p.23) : "[[...]] brackets equals:
\\ ((1-cq^n)(1-aba^{-n+k}/c)(1-q^{-n-1})(cq^n/ab)
\\ +(1-cq^n/a)(1-cq^n/b)(1-q^{-n+k-1}))
\\ = ((1-cq^n)(1-c/abq)(1-q^k) +\frac{cq^n}/{ab}(1-a)(1-b)(1-q^{-n+k-1}))."
\\ {} 7 vars with 4 terms highest degree 3 of total 7
{ id7_4_3_7a = +a*b*d*(a-c)*(a-g)*(a*e-b*f)
+b*(a-e)*(a*a-b*d)*(a*a*d-c*f*g)
-(a*a-b*d)*(a*a-b*f)*(a*b*d-c*e*g)
-a*(a*c-b*d)*(a*e-b*f)*(a*g-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 3 of total 7
{ id7_4_3_7b = +a*(b*c-d*e)*(c*g-e*f)*(b*f-d*g)
-a*(a*b-d*e)*(a*g-e*f)*(b*f-d*g)
+b*(a-c)*(a*g-e*f)*(a*b*f-c*d*g)
-g*(a-c)*(a*b-d*e)*(a*d*g-b*c*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 3 of total 7
{ id7_4_3_7c = +a*(a*d*f-c*e*g)*(a*d*f-c*e*g)
+b*(a*d*g+a*e*f+b*e*g)*(a*d*f-c*e*g)
+c*(a*d*g+a*e*f+b*e*g)*(a*d*g+a*e*f+b*e*g)
-a*(a*d*d+b*d*e+c*e*e)*(a*f*f+b*f*g+c*g*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011, (p. 25, equ. (8.3)):
\\ (1-aq^k)(1-b^k)(1-q^{-n-1+k}) -(1-cq^{k-1})(1-abq^{-n+k}/c)(1-q^k)
\\ = (abq^{-n+k}/c)[(1-cq^n)(1-abq^{-n+k}/c)(1-q^{-n-1})(cq^n/ab)
\\ +(1-cq^n/a)(1-cq^n/b)(1-q^{-n+k-1})].
\\ {} 7 vars with 4 terms highest degree 3 of total 8
{ id7_4_3_8a = +a*b*(a*b-c*e)*(a*f-c*d)*(a*g-c*d)
+b*(a*a-c*d)*(a*a-c*e)*(a*c*d-b*f*g)
-a*c*(a-b)*(a*e-b*d)*(a*c*d-b*f*g)
-c*d*(a*a-b*f)*(a*a-b*g)*(a*b-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 3 of total 8
{ id7_4_3_8b = +a*a*(a*d-c*e)*(a*g-e*f)*(c*g-d*f)
+a*c*e*(a-b)*(b*g-e*f)*(c*g-d*f)
-c*g*(a-b)*(a*d-c*e)*(a*a*g-b*e*f)
+(a*g-e*f)*(a*a*d-b*c*e)*(a*d*f-b*c*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 5
{ id7_4_4_5 = +a*d*g*(b-c)*(f-g) -a*(b*d-c*g)*(d*f-g*g) -c*g*g*(a-e)*(d-g)
+(d-g)*(a*b*d*f-c*e*g*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 6
{ id7_4_4_6a = +a*a*(a*d*d*e-b*c*c*f+a*b*b*g) -b*b*(b*d*d*d-c*c*c*e+a*a*a*g)
+c*c*(a*c*d*d-b*b*c*e+a*a*b*f) -d*d*(a*c*c*c-b*b*b*d+a*a*a*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 6
{ id7_4_4_6b = +a*a*(a*c*e*e-b*b*d*f+a*a*c*g) -a*c*(b*d*d*d-c*c*c*e+a*a*a*g)
+b*d*(a*c*d*d-b*b*c*e+a*a*b*f) -c*e*(a*c*c*c-b*b*b*d+a*a*a*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 7
{ id7_4_4_7a = +a*(a*c*e-b*d*f)*(a*c*e-b*d*f)
-b*g*(a*c*e-b*d*f)*(a*f+d*e+e*g)
+a*b*c*(a*f+d*e+e*g)*(a*f+d*e+e*g)
-(a*a*c+b*d*d+b*d*g)*(a*c*e*e+a*b*f*f+b*e*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 7
{ id7_4_4_7b = +a*(a*c*e-b*d*f)*(a*c*e-b*d*f)
-f*g*(a*c*e-b*d*f)*(a*d+b*e+e*g)
+a*c*f*(a*d+b*e+e*g)*(a*d+b*e+e*g)
-(a*a*c+b*b*f+b*f*g)*(a*c*e*e+a*d*d*f+d*e*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 7
{ id7_4_4_7c = +f*g*g*(a*d-b*e)*(a*d-b*e)
-c*f*(a*d-b*e)*(a*c*f+a*e*g+b*f*g)
+d*(a*c*f+a*e*g+b*f*g)*(a*c*f+a*e*g+b*f*g)
-(a*a*d*g+a*b*c*f+b*b*f*g)*(c*e*f+d*f*g+e*e*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 8
{ id7_4_4_8a = +a*a*(a*c*e-b*d*f)*(a*c*e-b*d*f)
+c*g*(a*c*e-b*d*f)*(a*a*f+a*d*e+d*f*g)
+b*c*(a*a*f+a*d*e+d*f*g)*(a*a*f+a*d*e+d*f*g)
-a*(a*a*c+b*d*d+c*d*g)*(a*b*f*f+a*c*e*e+c*e*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 8
{ id7_4_4_8b = +c*c*(a*c*e-b*d*f)*(a*c*e-b*d*f)
+a*g*(a*c*e-b*d*f)*(c*d*e+c*c*f+d*f*g)
+a*b*(c*d*e+c*c*f+d*f*g)*(c*d*e+c*c*f+d*f*g)
-c*(a*c*c+a*d*g+b*d*d)*(a*c*e*e+a*e*f*g+b*c*f*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 8
{ id7_4_4_8c = +b*b*(a*c*e-b*d*f)*(a*c*e-b*d*f)
+a*g*(a*c*e-b*d*f)*(b*c*f+b*d*e+d*f*g)
+a*b*(b*c*f+b*d*e+d*f*g)*(b*c*f+b*d*e+d*f*g)
-(a*b*c*c+a*c*d*g+b*b*d*d)*(a*b*e*e+a*e*f*g+b*b*f*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 8
{ id7_4_4_8d = +g*g*(a*c*e-b*d*f)*(a*c*e-b*d*f)
+e*e*(a*c*e-b*d*f)*(a*d*g+b*c*g+b*d*e)
+e*f*(a*d*g+b*c*g+b*d*e)*(a*d*g+b*c*g+b*d*e)
-(a*b*e*e+a*a*e*g+b*b*f*g)*(c*c*e*g+c*d*e*e+d*d*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 8
{ id7_4_4_8e = +g*g*(a*e*e-b*d*f)*(a*e*e-b*d*f)
+c*e*(a*e*e-b*d*f)*(a*d*g+b*c*d+b*e*g)
+e*f*(a*d*g+b*c*d+b*e*g)*(a*d*g+b*c*d+b*e*g)
-(a*a*e*g+a*b*c*e+b*b*f*g)*(c*d*e*e+d*d*f*g+e*e*e*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 8
{ id7_4_4_8f = +g*g*(a*c*e-b*d*f)*(a*c*e-b*d*f)
+c*d*(a*c*e-b*d*f)*(a*f*g+b*d*f+b*e*g)
+c*d*(a*f*g+b*d*f+b*e*g)*(a*f*g+b*d*f+b*e*g)
-(a*a*c*g+a*b*c*d+b*b*d*g)*(c*d*e*f+c*e*e*g+d*f*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 4 terms highest degree 4 of total 8
{ id7_4_4_8g = +a*a*c*c*d*d*e*f -a*b*c*d*(a+d)*(e+f)*g*g
+b*b*(a+d)*(a+d)*g*g*g*g
-(a*b*g*g+b*d*g*g-a*c*d*e)*(a*b*g*g+b*d*g*g-a*c*d*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 26, equ. (8.5)).
\\ Replace {a -> a/b, b -> b/c, c -> c/d, d -> d/e, e -> e/f, f -> f/g}.
\\ {} 7 vars with 4 terms highest degree 4 of total 16
{ id7_4_4_16 =
+a*a*b*b*(b-c)*(c-d)*(d-e)*(e-f)*(f-g)*(a*a*e-b*b*b)*(a*a*a*g-b*b*b*b)
-a*b*(a-b)*(e-f)*(f-g)*(a*d-b*b)*(a*e-b*c)*(a*c*e-b*b*d)*(a*a*a*g-b*b*b*b)
-(a-b)*(a*c-b*b)*(a*d-b*c)*(a*e-b*d)*(a*g-b*e)*(a*a*f-b*b*b)*(a*a*e*g-b*b*b*f)
+(a*c-b*b)*(a*d-b*c)*(a*e-b*d)*(a*f-b*e)*(a*g-b*f)*(a*a*e-b*b*b)*(a*a*g-b*b*b)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011. (p. 26, equ. (8.5)).
\\ Replace {a -> a/g, b -> b/g, c -> c/g, d -> d/g, e -> e/g, f -> f/g}.
\\ {} 7 vars with 4 terms highest degree 5 of total 15
{ id7_4_5_15 =
+a*a*(b-g)*(c-g)*(d-g)*(e-g)*(f-g)*(a*a*g-b*c*d)*(a*a*a*g*g-b*c*d*e*f)
+g*g*(a-b)*(a-c)*(a-d)*(a-e)*(a-f)*(a*a*g-b*c*d)*(a*a*g*g*g-b*c*d*e*f)
-a*(a-g)*(e-g)*(f-g)*(a*g-b*c)*(a*g-b*d)*(a*g-c*d)*(a*a*a*g*g-b*c*d*e*f)
-g*(a-g)*(a-b)*(a-c)*(a-d)*(a*g-e*f)*(a*a*g*g-b*c*d*e)*(a*a*g*g-b*c*d*f)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "In Praise of an Elementary Identity of Euler",
\\ arXiv:1102.0659v2 24 Feb 2011, (p. 26, equ. (8.6)).
\\ {} 7 vars with 4 terms highest degree 6 of total 19
{ id7_4_6_19 =
+a*a*d*(b-g)*(c-g)*(d-g)*(e-g)*(f-g)*(a*a*a*g*g-b*c*d*e*f)*(a*a*a*g*g*g-b*c*d*d*e*f)
+d*g*g*(a-b)*(a-c)*(a-d)*(a-e)*(a-f)*(a*a*g*g*g-b*c*d*e*f)*(a*a*a*g*g*g-b*c*d*d*e*f)
-(a-d)*(a-g)*(d-g)*(a*a*g*g-b*c*d*e)*(a*a*g*g-b*c*d*f)*(a*a*g*g-b*d*e*f)*(a*a*g*g-c*d*e*f)
-(a-g)*(a*g-b*d)*(a*g-c*d)*(a*g-d*e)*(a*g-d*f)*(a*a*g*g*g-b*c*d*e*f)*(a*a*a*g*g-b*c*d*e*f)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into four rectangles.
\\ {} 7 vars with 5 terms highest degree 1 of total 2 [NC]
{ id7_5_1_2a = +(a-b)*(f-g) +(a-b)*(d-f) +(b-c)*(d-e) +(b-c)*(e-g)
-(a-c)*(d-g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 1 of total 2 [NC]
{ id7_5_1_2b = +(a-b)*(f-g) +(a-b)*(d-f) +(b-c)*(d-e) -(a-b)*(e-g)
-(a-c)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J. W. L. Glaisher, On the deduction of series from infinite products,
\\ Messenger of Math., 2 (1873), p. 138. "to make use of the identities
\\ (1-a)(1-b)(1-c)... = 1-a-b(1-a)-c(1-a)(1-b)-... (2)"
\\ {} 7 vars with 5 terms highest degree 1 of total 3 [NC]
{ id7_5_1_3a = +(a-b)*(c-d)*(e-f) +(b-g)*(c-d)*(e-f) +(a-g)*(d-g)*(e-f)
+(a-g)*(c-g)*(f-g) -(a-g)*(c-g)*(e-g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 1 of total 3
{ id7_5_1_3b = +(a-g)*(b-g)*(c-g) -(d-g)*(e-g)*(f-g) -(a-d)*(e-g)*(f-g)
-(a-g)*(b-e)*(f-g) -(a-g)*(b-g)*(c-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 1 of total 3
{ id7_5_1_3c = +(a-g)*(b-f)*(d-e) +(a-g)*(c-d)*(e-f) -(a-g)*(c-e)*(d-f)
-(a-c)*(b-g)*(d-e) +(a-b)*(c-g)*(d-e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 1 of total 3
{ id7_5_1_3d = -(a-g)*(b-g)*(f-g) +(a-d)*(b-e)*(c-g) +(a-d)*(c-g)*(e-g)
-(a-g)*(b-g)*(c-f) +(b-g)*(c-g)*(d-g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 1 of total 6
{ id7_5_1_6a = +(a-d)*(a-g)*(a-f)*(d-g)*(d-f)*(g-f)
+(a-d)*(a-g)*(d-g)*(e-f)*(c-f)*(b+f)
+(a-d)*(a-f)*(d-f)*(e-g)*(g-c)*(b+g)
+(a-g)*(a-f)*(d-e)*(d-c)*(g-f)*(b+d)
-(a-e)*(a-c)*(d-g)*(d-f)*(g-f)*(a+b) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 1 of total 6
{ id7_5_1_6b = +(a-g)*(b-g)*(c-g)*(a-b)*(a-c)*(b-c)
-(d-g)*(e-g)*(f-g)*(a-b)*(a-c)*(b-c)
-(a-g)*(b-g)*(a-b)*(c-d)*(c-e)*(c-f)
+(a-g)*(c-g)*(a-c)*(b-d)*(b-e)*(b-f)
-(b-g)*(c-g)*(a-d)*(a-e)*(a-f)*(b-c) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 125, equ. (2)) :
\\ "The identity of Edmund Hlawka: [[..]]" (last term was missing).
\\ Alice Simon and Peter Volkmann, "On Two Geometric Inequalities", Annales
\\ Mathematicae Silesianae 9 (1995), pp. 137-140. (p. 138, equ. (6)).
\\ Note: intended use is a=||A||, b=||B||, c=||C||, d=||A+B+C||, e=||B+C||,
\\ f=||A+C||, g=||A+B|| for some vector space norm ||_||.
\\ {} 7 vars with 5 terms highest degree 2 of total 2
{ id7_5_2_2 = -(a+b+c+d-e-f-g)*(a+b+c+d) +(a+b-g)*(c+d-g) +(b+c-e)*(d+a-e)
+(c+a-f)*(b+d-f) +(a*a+b*b+c*c+d*d-e*e-f*f-g*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ MR2178993 Zhang, Yusen and Chen, Wei
\\ q-triplicate inverse series relations with applications
\\ Rocky Mountain J. Math 35 (2005), 1407-1427. (p. 417, equ. (4.1)): "[[...]]
\\ We begin this section by giving a useful formula $$ (4.1) (1-d)(1-ea)(1-fa)
\\ +d(1-a)(1-b)(1-c) = (1-a)(1-bd)(1-cd) +a(1-d)(1-e)(1-f), $$ where efa=bcd."
\\ {} 7 vars with 5 terms highest degree 3 of total 5
{ id7_5_3_5 = +a*g*(d-g)*(e-g)*(f-g) -d*g*(a-g)*(b-g)*(c-g)
+(a-g)*(d-g)*(a*e*f-b*c*d) +(a-g)*(b*d-g*g)*(c*d-g*g)
-(d-g)*(a*e-g*g)*(a*f-g*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 3 of total 8
{ id7_5_3_8 = +a*b*c*d*g*(d-g)*(e-g)*(f-g) -a*d*e*f*g*(a-g)*(b-g)*(c-g)
+a*e*f*(a-g)*(b*d-g*g)*(c*d-g*g) -b*c*d*(d-g)*(a*e-g*g)*(a*f-g*g)
+g*g*g*(a-g)*(d-g)*(a*e*f-b*c*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 5 terms highest degree 4 of total 7
{ id7_5_4_7 = -d*d*e*f*(b+c+d)*(a*c-b*d-c*d)
+d*e*f*(a*c-b*d+c*c)*(b*d-c*e-d*e)
-d*f*(c+d)*(b*d-c*e+d*d)*(c*e-d*f-e*f)
+(c+d)*(d+e)*(a*c*e-d*d*f+c*e*f)*(d*f-e*g-f*g)
+(c+d)*(d+e)*(e+f)*(a*c*e*g+c*e*f*g-b*d*d*f-d*d*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 6 terms highest degree 1 of total 2 [NC]
{ id7_6_1_2a = +(a-g)*(b-g) -(c-g)*(d-g) -(a-e)*(b-f)
+(c-e)*(d-f) -(b-d)*(e-g) -(a-c)*(f-g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 6 terms highest degree 1 of total 2 [NC]
{ id7_6_1_2b = -(a-g)*(b-g) +(e-g)*(f-g) +(a-g)*(b-d)
+(c-g)*(d-f) +(a-c)*(d-g) +(c-e)*(f-g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 6 terms highest degree 1 of total 5
{ id7_6_1_5 = +(a-c)*(b-c)*(b-d)*(e-g)*(f-g) -(a-c)*(b-f)*(b-g)*(e-g)*(f-g)
+(a-c)*(b-f)*(c-g)*(d-g)*(e-g) -(a-e)*(b-g)*(c-g)*(d-f)*(e-f)
-(a-g)*(b-g)*(c-e)*(d-f)*(f-g) +(b-g)*(c-e)*(c-g)*(d-f)*(e-g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 6 terms highest degree 1 of total 11
{ id7_6_1_11 =
+(a-b)*(a-c)*(a-d)*(a-e)*(b-c)*(b-d)*(b-e)*(c-d)*(c-e)*(d-e)*(f-g)
-(a-b)*(a-c)*(a-d)*(a-f)*(b-c)*(b-d)*(b-f)*(c-d)*(c-f)*(d-f)*(e-g)
+(a-b)*(a-c)*(a-e)*(a-f)*(b-c)*(b-e)*(b-f)*(c-e)*(c-f)*(d-g)*(e-f)
-(a-b)*(a-d)*(a-e)*(a-f)*(b-d)*(b-e)*(b-f)*(c-g)*(d-e)*(d-f)*(e-f)
+(a-c)*(a-d)*(a-e)*(a-f)*(b-g)*(c-d)*(c-e)*(c-f)*(d-e)*(d-f)*(e-f)
-(a-g)*(b-c)*(b-d)*(b-e)*(b-f)*(c-d)*(c-e)*(c-f)*(d-e)*(d-f)*(e-f)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into six rectangles.
\\ {} 7 vars with 6 terms highest degree 2 of total 2 [NC]
{ id7_6_2_2 = -(a*b-e*g) +a*(b-d) +c*(d-f) +e*(f-g) +(a-c)*d +(c-e)*f ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 6 terms highest degree 3 of total 7
{ id7_6_3_7 = +a*(b-f)*(c-g)*(d-e)*(a*b*c+e*f*g)
+a*b*f*(c-g)*(d-e)*(a*c-e*g)
+a*c*g*(b-f)*(d-e)*(a*b-e*f)
+a*e*(b-f)*(c-g)*(a*b*c-d*f*g)
+(a-d)*(a*b*c-e*f*g)*(a*b*c-e*f*g)
-(a*b-e*f)*(a*c-e*g)*(a*b*c-d*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 7 vars with 6 terms highest degree 3 of total 8
{ id7_6_3_8 = -a*d*(b-e)*(c-g)*(f-g)*(a*b*c+d*e*f)
-a*b*d*e*(c-g)*(f-g)*(a*c-d*f)
+a*d*f*(b-e)*(c-g)*(a*b*c-d*e*g)
-a*c*d*(b-e)*(f-g)*(a*b*g-d*e*f)
+g*(a-d)*(a*b*c-d*e*f)*(a*b*c-d*e*f)
-(a*c-d*f)*(a*b*c-d*e*g)*(a*b*g-d*e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the Krattenthaler's Determinant Formula case n=3 available at
\\ "http://dlmf.nist.gov/1.3#E16" and other sources.
\\ {} 7 vars with 7 terms highest degree 1 of total 6 [TS]
{ id7_7_1_6 = +(a-b)*(a-c)*(b-c)*(d-f)*(e-f)*(e-g)
-(a-d)*(a-e)*(b-e)*(b-f)*(c-f)*(c-g)
+(a-d)*(a-e)*(b-f)*(b-g)*(c-e)*(c-f)
+(a-e)*(a-f)*(b-d)*(b-e)*(c-f)*(c-g)
-(a-e)*(a-f)*(b-f)*(b-g)*(c-d)*(c-e)
+(a-f)*(a-g)*(b-e)*(b-f)*(c-d)*(c-e)
-(a-f)*(a-g)*(b-d)*(b-e)*(c-e)*(c-f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ T. Amdeberhan, O. Espinosa, V. Moll, A. Straub, "Wallis-Ramanujan-Schur-
\\ Feynman", (p.5, equ. (4.3)) case n=6 : "\prod_{k=1}^n \frac{1}{y+b_k} =
\\ \sum_{k=1}^n \frac{1}{y+b_k}\prod_{j=1,j\ne k}^n\frac{1}{b_j-b_k}. "
\\ Marc Chamberland and Doron Zeilberger, "A Short Proof of a Ptolemy-Like
\\ Relation for an Even number of Points on a Circle Discovered by Jane
\\ McDougall", equation (Joseph') case N=7, r=0 : "$$ 0 = \sum_{i=1}^N
\\ \frac{z_i^r} {(z_i-z_1)\dots(z_i-z_{i-1})(z_i-z_{i+1})\dots(z_i-z_N)}. $$"
\\ {} 7 vars with 7 terms highest degree 1 of total 15
{ id7_7_1_15 =
+(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e)*(a-f)*(b-f)*(c-f)*(d-f)*(e-f)
-(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-e)*(b-e)*(c-e)*(d-e)*(a-g)*(b-g)*(c-g)*(d-g)*(e-g)
+(a-b)*(a-c)*(b-c)*(a-d)*(b-d)*(c-d)*(a-f)*(b-f)*(c-f)*(d-f)*(a-g)*(b-g)*(c-g)*(d-g)*(f-g)
-(a-b)*(a-c)*(b-c)*(a-e)*(b-e)*(c-e)*(a-f)*(b-f)*(c-f)*(e-f)*(a-g)*(b-g)*(c-g)*(e-g)*(f-g)
+(a-b)*(a-d)*(b-d)*(a-e)*(b-e)*(d-e)*(a-f)*(b-f)*(d-f)*(e-f)*(a-g)*(b-g)*(d-g)*(e-g)*(f-g)
-(a-c)*(a-d)*(c-d)*(a-e)*(c-e)*(d-e)*(a-f)*(c-f)*(d-f)*(e-f)*(a-g)*(c-g)*(d-g)*(e-g)*(f-g)
+(b-c)*(b-d)*(c-d)*(b-e)*(c-e)*(d-e)*(b-f)*(c-f)*(d-f)*(e-f)*(b-g)*(c-g)*(d-g)*(e-g)*(f-g)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Eduardo Casas-Alvero, Analytic Projective Geometry, 2014, (p. 25) :
\\ "Since $$ \lambda\mu(u+v+w)-(u+\lambda v+\mu w)-(\lambda\mu-1)u-\lambda
\\ (\mu-1)v-\mu(\lambda-1)w = 0, $$ the claim is proved."
\\ Replace {\lambda -> a/f, \mu -> b/g, u -> c, v -> d, w -> e}.
\\ {} 7 vars with 7 terms highest degree 2 of total 3
{ id7_7_2_3 = +a*d*g +b*e*f +c*f*g +a*d*(b-g) +b*e*(a-f) +c*(a*b-f*g)
-a*b*(c+d+e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Math Stack Exchange answer to question 591078. The answer: "Note that
\\ (|a|+|b|+|c|-|b+c|-|a+c|-|a+b|+|a+b+c|)(|a|+|b|+|c|+|a+b+c|) =
\\ (|b|+|c|-|b+c|)(|a|-|b+c|+|a+b+c|) +(|c|+|a|-|c+a|)(|b|-|c+a|+|a+b+c|)
\\ +(|a|+|b|-|a+b|)(|c|-|a+b|+|a+b+c|). By done!" Replace {|a| -> a,
\\ |b| -> b, |c| -> c, |a+b| -> d, |a+c| -> e, |b+c| -> f, |a+b+c| -> g}.
\\ This is id7_5_2_2 with its last term split into 7 terms.
\\ {} 7 vars with 11 terms highest degree 1 of total 2
{ id7_11_1_2 = -(a+b+c+d-e-f-g)*(a+b+c+d) +(a+b-g)*(c+d-g) +(b+c-e)*(d+a-e)
+(c+a-f)*(b+d-f) +a*a +b*b +c*c +d*d -e*e -f*f -g*g ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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\\ 8 variables \\
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\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the homogeneous Plucker identity for line coordinates in 3 space as in
\\ "https://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding#Pl%C3%BCcker_relations".
\\ case n=2 of det(A*B) = det(A)*det(B) for nXn matrices.
\\ Carl F. Gauss, Disquisitiones Arithmeticae, art. 159 : " [[...]] (\alpha
\\ \alpha'+\beta\gamma')(\gamma\beta'+\delta\delta')-(\alpha\beta'+\beta
\\ \delta')(\gamma\alpha'+\delta\gamma')=(\alpha\delta-\beta\gamma)(\alpha'
\\ \delta'-\beta'\gamma')"
\\ Theodore S. Motzkin, The pentagon in the projective plane, with a comment
\\ on Napier's rule, Bull. Amer. Math. Soc. 51, (1945), 985-989. (p. 986):
\\ "[[...]] . Denoting 1:[b:c,d:e] by \bar{a}, and cyclically, we obtain
\\ \bar{a}\bar{c}=1-\bar{b}." (note: [b:c,d:e] is a cross-ratio)
\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 5): "... we obtain
\\ (1-cq^k)(a-b) + (1-aq^k)(b-c) + (1-bq^k)(c-a) = 0."
\\ {} 8 vars with 3 terms highest degree 2 of total 4
{ id8_3_2_4a = +(a*f-b*e)*(c*h-d*g) -(a*g-c*e)*(b*h-d*f)
+(a*h-d*e)*(b*g-c*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 2 of total 4
{ id8_3_2_4b = -f*(d-e)*(a*g-a*h+b*h-b*f+c*f-c*g)
+(a*h-c*f+d*f-d*h)*(a*g-b*f-e*g+e*f)
-(a*g-b*f-d*g+d*f)*(a*h-c*f-e*h+e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gasper+Rahman, Basic Hypergeometric Series 2nd ed. (p. 303):
\\ "[[...]] which is simply the elliptic analogue of the trivial identity
\\ $$ (1-x\lambda)(1-x/\lambda)(1-\mu\nu)(1-\mu/\nu)
\\ -(1-x\nu)(1-x/\nu)(1-\lambda\mu)(1-\mu/\lambda) = \frac{\mu,\lambda}
\\ (1-x\mu)(1-x/\mu)(1-\lambda\nu)(1-\lambda/\nu). (11.1.1) $$"
\\ {} 8 vars with 3 terms highest degree 2 of total 8
{ id8_3_2_8 = +(a*c-b*d)*(a*d-b*c)*(e*g-f*h)*(e*h-f*g)
-(a*e-b*f)*(a*f-b*e)*(c*g-d*h)*(c*h-d*g)
+(a*g-b*h)*(a*h-b*g)*(c*e-d*f)*(c*f-d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 3 of total 4
{ id8_3_3_4a = +(a*g-c*e)*(b*f-d*h) -(b-d)*(a*g*h-c*e*f)
-(f-h)*(a*b*g-c*d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 3 of total 4
{ id8_3_3_4b = +a*(b*e*h-c*d*f) -b*(a*e*h+b*c*g) +c*(a*d*f+b*b*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 3 of total 4
{ id8_3_3_4c = +a*(b*e*h-c*c*f) -b*(a*e*h+c*d*g) +c*(a*c*f+b*d*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 3 of total 5
{ id8_3_3_5 = +(a*c-b*b)*(d*e*g-f*f*h) +(a*d*g-b*f*h)*(b*f-c*e)
-(a*f-b*e)*(b*d*g-c*f*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 7): ".... This leads to
\\ (1-cq^k/a)(1-cq^k/b) - c/ad(1-aq^k)(1-b^k) = (1-c/ab)(1-cq^{2k})."
\\ {} 8 vars with 3 terms highest degree 3 of total 6
{ id8_3_3_6 = +a*e*(b*d-f*h)*(c*d-g*h) +(a*d*d-e*h*h)*(a*f*g-b*c*e)
-(a*d*f-b*e*h)*(a*d*g-c*e*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 5): "... we finally get
\\ (1-aq^k)(1-cq^k/a) - (1-bq^k)(1-cq^k/b) = q^k(b-a)(1-c/ab)."
\\ {} 8 vars with 3 terms highest degree 3 of total 7
{ id8_3_3_7 = +a*e*(b*d-f*h)*(b*g*h-c*d*f) -b*f*(a*d-e*h)*(a*g*h-c*d*e)
+d*h*(a*f-b*e)*(a*b*g-c*e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 8): "... this yields
\\ (1-A)(1-C/A)(1-BD)(1-CD/B) - (1-B)(1-C/B)(1-AD)(1-CD/A)
\\ = (B-A)(1-C/AB)(1-D)(1-CD). (4.1)"
\\ {} 8 vars with 3 terms highest degree 3 of total 8
{ id8_3_3_8a = +(a-e)*(a*g-c*e)*(b*d-f*h)*(b*g*h-c*d*f)
-(b-f)*(a*d-e*h)*(b*g-c*f)*(a*g*h-c*d*e)
+(d-h)*(a*f-b*e)*(c*d-g*h)*(a*b*g-c*e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 3 of total 8
{ id8_3_3_8b = +(a-e)*(d-h)*(a*b*g-c*e*f)*(b*c*h-d*f*g)
-(b-f)*(c-g)*(a*b*h-d*e*f)*(a*d*g-c*e*h)
+(a*b-e*f)*(a*g-c*e)*(b*h-d*f)*(c*h-d*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 3 of total 8
{ id8_3_3_8c = +(b-f)*(a*f-b*e)*(c*h-d*g)*(a*g*h-c*d*e)
-(c-g)*(a*g-c*e)*(b*h-d*f)*(a*f*h-b*d*e)
+(d-h)*(a*h-d*e)*(b*g-c*f)*(a*f*g-b*c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 3 of total 8
{ id8_3_3_8d = +(a-e)*(d-h)*(a*c*f-b*e*g)*(b*c*d-f*g*h)
+(b-f)*(c-g)*(a*c*d-e*g*h)*(a*f*h-b*d*e)
-(a*c-e*g)*(a*f-b*e)*(b*d-f*h)*(c*d-g*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 4 of total 7
{ id8_3_4_7a = -(a*d-b*c)*(b*c*f-d*e*h)*g*h
+(a*e*g-b*b*h)*(c*c*f*g-d*d*h*h)
+(a*c*f*g-b*d*h*h)*(b*d*h-c*e*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 4 of total 7
{ id8_3_4_7b = -(a*d-b*c)*(a*b*e-c*d*g)*f*h
+a*c*(f-h)*(b*b*e*f-d*d*g*h)
+(a*b*e*f-c*d*g*h)*(a*d*h-b*c*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 4 of total 8
{ id8_3_4_8a = -(a-e)*(b*d-f*h)*(b*c-f*g)*(a*c*d-e*g*h)
+(b-f)*(a*c-e*g)*(a*d-e*h)*(b*c*d-f*g*h)
+(c-g)*(a*f-b*e)*(d-h)*(a*b*c*d-e*f*g*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 2, (p. 407) :
\\ "L. Euler^{48} noted that (a\alpha p^2+b\beta q^2)(abr^2+\alpha\beta s^2)
\\ = \alpha b(apr\pm\beta qs)^2 + a\beta(\alpha ps\mp bqr)^2. ..."
\\ Replace {\alpha -> -c, \beta -> d, p -> e, q -> f, r -> g, s -> -h}.
\\ Leonard E. Dickson, History of the Theory of Numbers, Vol. 3, (p. 60) :
\\ "A. J. Lexell noted ..." the same equation as Euler in Vol.2, (p. 407).
\\ {} 8 vars with 3 terms highest degree 4 of total 8
{ id8_3_4_8b = -(a*c*e*e-b*d*f*f)*(a*b*g*g-c*d*h*h)
+b*c*(a*e*g-d*f*h)*(a*e*g-d*f*h)
-a*d*(c*e*h-b*f*g)*(c*e*h-b*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 4 of total 16
{ id8_3_4_16 =
-(a*b*c*d-e*f*g*h)*(a*b*g*h-c*d*e*f)*(a*c*f*h-b*d*e*g)*(a*d*f*g-b*c*e*h)
+(a*b*c*h-d*e*f*g)*(a*b*d*g-c*e*f*h)*(a*c*d*f-b*e*g*h)*(a*f*g*h-b*c*d*e)
+a*b*c*d*e*f*g*h*(a-e)*(a+e)*(b-f)*(b+f)*(c-g)*(c+g)*(d-h)*(d+h)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 5 of total 6
{ id8_3_5_6 = +a*(b*b*d*g*h+b*c*c*g*h+b*d*d*e*h+c*c*d*e*h+c*d*d*f*g)
-d*(a*a*c*g*h+a*b*b*g*h+a*c*c*e*h+b*b*c*e*h+b*c*c*f*g)
+(a*d-b*c)*(a*c*g*h-b*d*e*h-c*d*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 3 terms highest degree 5 of total 10
{ id8_3_5_10 = +a*b*(c-f)*(d-g)*(e-h)*(a*a*f*g*h-b*b*c*d*e)
-(a-b)*(a*f*g-b*c*d)*(a*f*h-b*c*e)*(a*g*h-b*d*e)
+(a*f-b*c)*(a*g-b*d)*(a*h-b*e)*(a*f*g*h-b*c*d*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 4 terms highest degree 2 of total 3
{ id8_4_2_3 = +a*(c*h-e*f) -b*(c*g-d*f) -c*(a*h-b*g) +f*(a*e-b*d) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 8 vars with 4 terms highest degree 2 of total 4
{ id8_4_2_4a = +(a-b)*(c-d)*(e*g-f*h) -(a-b)*(c*e-d*f)*(g-h)
-(c-d)*(e-f)*(a*g-b*h) +(a*c-b*d)*(e-f)*(g-h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ J. M. Steele, The Cauchy-Schwarz Master Class, 2004 (p. 49) case n=2 :
\\ "Exercise 3.7 (The Four-Letter Identity via Polarization) [[...]] $$
\\ \sum_{i=1}^na_js_j\sum_{i=1}^nb_jt_j-\sum_{i=1}^na_jb_j\sum_{i=1}^ns_jt_j
\\ = \sum_{1\le j -a, w_2 -> a-c, w_3 -> c-e, w_4 -> g, a_1 -> b, a_2 ->d,
\\ a_3 -> f, a_4 ->h}.
\\ {} 8 vars with 8 terms highest degree 1 of total 2 [NC]
{ id8_8_1_2 = -a*b +g*h +a*(b-d) +c*(d-f) +e*(f-h)
+(a-c)*d +(c-e)*f +(e-g)*h ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 16) case n=4 :
\\ "Aczel's Inequality. If $a,b$ are real n-tuples with $a_1^2-\sum_{i=2}^n
\\ a_i^2>0$ then $$ (a_1^2-\sum_{i=2}^n a_i^2)(b_1^2-\sum_{i=2}^n b_i^2)\le
\\ (a_1b_1-\sum_{i=2}^n a_ib_i)^2 $$, with equality if and only if $a~b$."
\\ {} 8 vars with 8 terms highest degree 2 of total 4
{ id8_8_2_4 = +(a*a-b*b-c*c-d*d)*(e*e-f*f-g*g-h*h)
-(a*e-b*f-c*g-d*h)*(a*e-b*f-c*g-d*h) +(a*f-b*e)*(a*f-b*e)
+(a*g-c*e)*(a*g-c*e) +(a*h-d*e)*(a*h-d*e) -(b*h-d*f)*(b*h-d*f)
-(b*g-c*f)*(b*g-c*f) -(c*h-d*g)*(c*h-d*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 9 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {i -> 0} to get id8_3_2_4b.
\\ {} 9 vars with 3 terms highest degree 2 of total 4
{ id9_3_2_4 = +(a*g-a*h+b*h-b*f+c*f-c*g)*(a*i-d*f-e*i+e*f)
+(a*h-a*i+c*i-c*f+d*f-d*h)*(a*g-b*f-e*g+e*f)
+(a*i-a*g+b*f-b*i+d*g-d*f)*(a*h-c*f-e*h+e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is an example of the Desananot-Jacobi identity for a 4x4 matrix.
\\ {} 9 vars with 3 terms highest degree 3 of total 4
{ id9_3_3_4 = +(a*e*i-a*h*f+b*f*g-b*d*i+c*d*h-c*e*g)*i
-(a*i-c*g)*(e*i-f*h) +(b*i-c*h)*(d*i-f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 3 terms highest degree 4 of total 7
{ id9_3_4_7 = +a*b*(c*e-d*f)*(d*f*h-e*g*i)
+(a*f*i-b*d*e)*(a*c*f*h-b*d*e*g)
-(a*c*i-b*d*d)*(a*f*f*h-b*e*e*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 3 terms highest degree 5 of total 6
{ id9_3_5_6 = +h*(b*e*e*f*f+a*d*f*f*f+a*e*e*e*g+c*d*d*g*g-d*e*f*g*i)
-d*(b*f*f*g*g+a*e*g*g*g+a*f*f*f*h+c*e*e*h*h-e*f*g*h*i)
+(b*f*f+a*e*g-c*d*h)*(d*g*g-e*e*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 4 terms highest degree 2 of total 3
{ id9_4_2_3a = +a*(b*c-d*e+f*g) -b*(a*c-e*i+g*h)
+e*(a*d-b*i+c*g) -g*(a*f-b*h+c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Remove c*e*g from last two terms of id9_4_2_3a.
\\ In a proof of A(n+1)*B(n)-A(n)*B(n+1) = (-1)^n where A(n)/B(n)
\\ are the convergents of a simple continued fraction by induction.
\\ From the vector triple product Ax(BxC) = (A.C)B - (A.B)C where
\\ A=(-e,-g,-c), B=(i,-h,-a), C=(-d,f,b).
\\ {} 9 vars with 4 terms highest degree 2 of total 3
{ id9_4_2_3b = +a*(b*c-d*e+f*g) -b*(a*c-e*i+g*h)
+e*(a*d-b*i) -g*(a*f-b*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 4 terms highest degree 3 of total 6
{ id9_4_3_6 = -(a-d)*(a-g)*(d-g)*(b*e*h-c*f*i)
+(a-d)*(h-i)*(a*b-c*g)*(d*e-f*g)
+(b-c)*(d-g)*(a*f-d*e)*(a*i-g*h)
+(a-g)*(e-f)*(a*b-c*d)*(d*i-g*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "A Short Proof of an Identity of Sylvester", Internat. J.
\\ Math. & Math. Sci., v. 22, n. 2 (1999) 431-435, (p. 432, equ. 2.1) case n=3
\\ of an identity due to Milne: \sum_{r=1}^n (1-y_r) \prod_{i=1,i\ne r}^n
\\ \frac{1-x_iy_i/x_r}{1-x_i/x_r} = 1-y_1y_2\cdots y_n.
\\ {} 9 vars with 4 terms highest degree 3 of total 9
{ id9_4_3_9 = +(a*f-c*d)*(a*i-c*g)*(d*i-f*g)*(b*e*h-c*f*i)
-(h-i)*(a*f-c*d)*(a*b*i-c*c*g)*(d*e*i-f*f*g)
-(b-c)*(d*i-f*g)*(a*f*f-c*d*e)*(a*i*i-c*g*h)
-(e-f)*(a*i-c*g)*(a*b*f-c*c*d)*(d*i*i-f*g*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 4 terms highest degree 4 of total 7
{ id9_4_4_7 = +a*b*c*(d*f-e*g)*(c*f-h*i)
-a*b*d*(c*f-h*i)*(c*f-h*i)
-(a*e*f-b*i*i)*(a*d*h*h-b*c*c*g)
+(a*e*h-b*c*i)*(a*d*f*h-b*c*g*i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ From Tito Piezas III site "https://sites.google.com/site/tpiezas/005b"
\\ section 5 about V. Arnold's Perfect Forms, Theorem 1.
\\ {} 9 vars with 4 terms highest degree 4 of total 9
{ id9_4_4_9 =
-(a*h*h+b*h*i+c*i*i)*(a*f*f+b*f*g+c*g*g)*(a*d*d+b*d*e+c*e*e)
+a*(-h*g*c*e+i*c*f*e+g*i*c*d+a*h*f*d+b*i*f*d)*(-h*g*c*e+i*c*f*e+g*i*c*d+a*h*f*d+b*i*f*d)
+b*(-h*g*c*e+i*c*f*e+g*i*c*d+a*h*f*d+b*i*f*d)*(+b*h*g*e+g*i*c*e+a*h*f*e+a*h*g*d-a*i*f*d)
+c*(+b*h*g*e+g*i*c*e+a*h*f*e+a*h*g*d-a*i*f*d)*(+b*h*g*e+g*i*c*e+a*h*f*e+a*h*g*d-a*i*f*d)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ case n=3 of 0 = (x_1-y_1)(x_2-y_2)...(x_n-y_n) +
\\ (y_1-z_1)*(x_2-y_2)...(x_n-y_n) + (x_1-z_1)*(y_2-z_2)*(x_3-y_3)...(x_n-y_n)
\\ + ... + (x_1-z_1)*(x_2-z_2)...(y_n-z_n) - (x_1-z_1)*(x_2-z_2)...(x_n-z_n).
\\ {} 9 vars with 5 terms highest degree 1 of total 3 [NC]
{ id9_5_1_3 = +(a-b)*(d-e)*(g-h) +(b-c)*(d-e)*(g-h) +(a-c)*(e-f)*(g-h)
+(a-c)*(d-f)*(h-i) -(a-c)*(d-f)*(g-i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Dodgson condensation of 3x3 determinant in terms of 1x1 and 2x2 determinants
\\ {} 9 vars with 5 terms highest degree 2 of total 4
{ id9_5_2_4a = +(a*e-b*d)*(e*i-f*h) -(b*f-c*e)*(d*h-e*g)
-a*e*(e*i-f*h) -d*e*(c*h-b*i) -g*e*(b*f-c*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Matjaz Konvalinka, "Non-commutative Sylvester's Determinant Identity",
\\ arXiv:math/0703213, p. 1, example 1.2 case n = 1, m = 3.
\\ {} 9 vars with 5 terms highest degree 2 of total 4
{ id9_5_2_4b = +(a*e-b*d)*(e*i-f*h) -(b*f-c*e)*(d*h-e*g)
+d*e*(b*i-c*h) -e*e*(a*i-c*g) +e*f*(a*h-b*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {i -> 0} to get id8_6_1_4.
\\ {} 9 vars with 6 terms highest degree 1 of total 4 [NC]
{ id9_6_1_4a = +(a-i)*(b-i)*(c-i)*(d-i) -(e-i)*(f-i)*(g-i)*(h-i)
-(a-e)*(f-i)*(g-i)*(h-i) -(a-i)*(b-f)*(g-i)*(h-i)
-(a-i)*(b-i)*(c-g)*(h-i) -(a-i)*(b-i)*(c-i)*(d-h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 6 terms highest degree 1 of total 4
{ id9_6_1_4b = +(a-i)*(b-i)*(c-i)*(d-i) -(a-i)*(b-i)*(c-i)*(h-i)
-(a-i)*(b-i)*(d-h)*(g-i) -(a-i)*(c-g)*(d-h)*(f-i)
-(b-f)*(c-g)*(d-h)*(e-i) -(a-e)*(b-f)*(c-g)*(d-h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 6 terms highest degree 1 of total 10
{ id9_6_1_10 =
+(a-i)*(b-i)*(c-i)*(d-i)*(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d)
-(e-i)*(f-i)*(g-i)*(h-i)*(a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d)
+(a-i)*(b-i)*(c-i)*(a-b)*(a-c)*(b-c)*(d-e)*(d-f)*(d-g)*(d-h)
-(a-i)*(b-i)*(d-i)*(a-b)*(a-d)*(b-d)*(c-e)*(c-f)*(c-g)*(c-h)
+(a-i)*(c-i)*(d-i)*(a-c)*(a-d)*(b-e)*(b-f)*(b-g)*(b-h)*(c-d)
-(b-i)*(c-i)*(d-i)*(a-e)*(a-f)*(a-g)*(a-h)*(b-c)*(b-d)*(c-d)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {i -> 0} to get id8_8_1_2.
\\ {} 9 vars with 8 terms highest degree 1 of total 2 [NC]
{ id9_8_1_2 = -(a-i)*(b-i) +(g-i)*(h-i) +(a-i)*(b-d) +(c-i)*(d-f)
+(e-i)*(f-h) +(a-c)*(d-i) +(c-e)*(f-i) +(e-g)*(h-i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into seven rectangles.
\\ {} 9 vars with 8 terms highest degree 2 of total 2 [NC]
{ id9_8_2_2 = -(a*b-g*i) +a*(b-d) +c*(d-f) +e*(f-h) +g*(h-i)
+(a-c)*d +(c-e)*f +(e-g)*h ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 9 vars with 11 terms highest degree 4 of total 7
{ id9_11_4_7 =
-(a-d)*(a+d)*(a-e)*(a-h)*(d-e)*(d-h)*(f-g)
-(a-e)*(a-h)*(b-d)*(d-e)*(d-h)*(e*f-g*h)
-(a-e)*(a-h)*(d-e)*(d-i)*(d+i)*(c*h-d*g)
+(a-e)*(a-h)*(d-h)*(d-i)*(d+i)*(c*e-d*f)
-(a-e)*(a-i)*(a+i)*(d-e)*(d-h)*(a*g-b*h)
+(a-e)*(a-h)*(d-e)*(d-h)*(a*a*f+a*g*h+b*e*f-b*g*h-b*i*i-f*i*i)
-(a-h)*(d-h)*(d-e)*(a*b-e*f)*(a*e-i*i)
+(a-h)*(d-e)*(e-h)*(a*a*b*d-a*c*d*d+a*c*i*i-b*d*i*i)
+(a-d)*(a-h)*(d-e)*(e*e*f*h-e*g*h*h+e*g*i*i-f*h*i*i)
-(a-d)*(a-h)*(e-h)*(c*d*d*e-d*e*e*f-c*e*i*i+d*f*i*i)
-(a-d)*(d-e)*(e-h)*(a*a*b*h-a*g*h*h+a*g*i*i-b*h*i*i)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 10 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Donald E. Knuth, Axioms and Hulls, Springer 1992, (p.56 equation (14.3)) :
\\ "... $$ |tpq||tsr|+|tqr||tsp|+|trp||tsq|=0. $$"
\\ {} 10 vars with 3 terms highest degree 2 of total 4
{ id10_3_2_4 = +(a*g-a*h+b*h-b*f+c*f-c*g)*(a*j-a*i+d*f-d*j+e*i-e*f)
+(a*h-a*i+c*i-c*f+d*f-d*h)*(a*j-a*g+b*f-b*j+e*g-e*f)
+(a*i-a*g+b*f-b*i+d*g-d*f)*(a*j-a*h+c*f-c*j+e*h-e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Christian Krattenthaler, A Systematic List of Two- and Three-term Contiguous
\\ Relations for Basic Hypergeometric Series, (p. 10): "... This results into
\\ (1-aq^k)(1-cq^k/a)(1-b/d)(1-c/bd) - (1-bq^k)(1-cq^k/b)(1-a/d)(1-c/ad)
\\ = \frac{1}{d}(a-b)(1-c/ab)(1-q^kd)(1-cq^k/d)."
\\ {} 10 vars with 3 terms highest degree 3 of total 10
{ id10_3_3_10 = +(a*e-f*j)*(b*i-d*g)*(a*h*j-c*e*f)*(b*d*h-c*g*i)
-(a*i-d*f)*(b*e-g*j)*(a*d*h-c*f*i)*(b*h*j-c*e*g)
-(a*g-b*f)*(d*e-i*j)*(a*b*h-c*f*g)*(c*e*i-d*h*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 107) : "Guha's
\\ Inequality. If $p\ge q>0, x\ge y>0$ then $(px+y+a)(x+qy+a)\ge[(p+1)x+a]
\\ [(q+1)y+a].$ [[...]] the difference [[...]] is just $(px-qy)(x-y)$."
\\ {} 10 vars with 3 terms highest degree 4 of total 8
{ id10_3_4_8 = +a*a*(d*j-e*i)*(b*d*h*j-c*e*g*i)
-(a*d*h*j+a*c*e*i+c*d*e*f)*(a*b*d*j+a*e*g*i+b*d*e*f)
+d*e*(a*b*i+a*g*i+b*d*f)*(a*c*j+a*h*j+c*e*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 3 terms highest degree 4 of total 10
{ id10_3_4_10 = +(a*g-b*f)*(c*i-d*h)*(c*j-e*h)*(a*b*i*j-d*e*f*g)
-(a*h-c*f)*(b*i-d*g)*(b*j-e*g)*(a*c*i*j-d*e*f*h)
+(a*i-d*f)*(a*j-e*f)*(b*h-c*g)*(b*c*i*j-d*e*g*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gasper+Rahman, Basic Hypergeometric Series 2nd ed. (p. 321, equ. (11.4.3)):
\\ "[[...]] (11.4.2) can be rewritten as $$ \theta(aq/b,aq/c,aq/d,aq/bcd;p) +
\\ \theta(b,c,d,a^2q^2/bcd;p)aq/bcd = \theta(aq,aq/bc,aq/bd,aq/cd;p). $$"
\\ {} 10 vars with 3 terms highest degree 7 of total 14
{ id10_3_7_14 =
+a*e*f*j*(b-g)*(c-h)*(d-i)*(a*a*e*e*g*h*i-b*c*d*f*f*j*j)
-(a*e-f*j)*(a*e*g*h-b*c*f*j)*(a*e*g*i-b*d*f*j)*(a*e*h*i-c*d*f*j)
+(a*e*g-b*f*j)*(a*e*h-c*f*j)*(a*e*i-d*f*j)*(a*e*g*h*i-b*c*d*f*j)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 4 terms highest degree 2 of total 3
{ id10_4_2_3 = +a*(b*c+d*e-f*g) -c*(a*b+f*h-e*j)
-e*(a*d+c*j-f*i) +f*(a*g+c*h-e*i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 4 terms highest degree 3 of total 4
{ id10_4_3_4 = +a*(b*g*j-d*f*h) -b*(a*g*j-c*f*i)
-c*(b*f*i-d*e*j) +d*(a*f*h-c*e*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The Lyness 5-cycle: If a(n+2) = (a(n+1)+1)/a(n) for n=1,2,3,4, then
\\ a(6)=a(1) if all a(n) are non-zero. One proof uses the algebraic identity
\\ 0 = (a2*a4-a3-1) -(a3*a5-a4-1) +a4*(a1*a3-a2-1) -a3*(a4*a1-a5-1). Now
\\ replace {a1 -> f/a, a2 -> g/b, a3 -> h/c, a4 -> i/d, a5 -> j/e}.
\\ {} 10 vars with 4 terms highest degree 3 of total 5
{ id10_4_3_5 = +a*b*(c*d*e+c*i*e-d*j*h) -a*e*(b*c*d+b*d*h-c*g*i)
+b*h*(a*d*e+a*d*j-e*f*i) -e*i*(a*b*c+a*c*g-b*f*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ A. Berkovich and K. Grizzell, "On the Class of Dominant and Subordinate
\\ Products. (p. 3) : "... since the identity $$ (1-t\alpha)(1-t\beta)(1-txy)
\\ -(1-tx)(1-ty)(1-t\alpha\beta)=t(x-\alpha)(1-\beta)(1-ty)+t(y-\beta)
\\ (1-t\alpha)(1-x) $$ is true, ..."
\\ {} 10 vars with 4 terms highest degree 3 of total 7
{ id10_4_3_7 = +a*b*(a*e-b*f)*(d*g-c*h)*(i-j) +a*b*(a*g-b*h)*(e*i-f*j)*(c-d)
+(a*c*e-b*d*f)*(a*g-b*h)*(a*j-b*i)
-(a*c-b*d)*(a*e-b*f)*(a*g*j-b*h*i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 4 terms highest degree 4 of total 5
{ id10_4_4_5a = +b*d*e*(g*i+h*j) +g*(a*c*f*j-b*d*e*i-b*d*f*j)
-c*f*j*(a*g-b*h) -b*j*(c*f*h+d*e*h-d*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 4 terms highest degree 4 of total 5
{ id10_4_4_5b = +f*g*i*(a*c-b*d) -f*(a*g-b*h)*(c*i-d*j)
-d*(a*f*g*j-b*e*h*i-b*f*h*j) -b*i*(c*f*h+d*e*h-d*f*g) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ MR2178993 Zhang, Yusen and Chen, Wei
\\ q-triplicate inverse series relations with applications
\\ Rocky Mountain J. Math 35 (2005), 1407-1427. (p. 1417): "[[...]]
\\ We begin this section by giving a useful formula $$ (4.1) (1-d)(1-ea)(1-fa)
\\ +d(1-a)(1-b)(1-c) = (1-a)(1-bd)(1-cd) +a(1-d)(1-e)(1-f), $$ where efa=bcd."
\\ {} 10 vars with 4 terms highest degree 5 of total 7
{ id10_4_5_7a = +g*h*i*j*(a-b)*(c-d)*(e-f)
+(g-h)*(i-j)*(a*d*f*h*i-b*c*e*g*j)
+(g-h)*(a*i-b*j)*(c*e*g*j-d*f*h*i)
-i*j*(a-b)*(c*g-d*h)*(e*g-f*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 4 terms highest degree 5 of total 7
{ id10_4_5_7b = +a*b*c*d*(e-f)*(g-h)*(i-j)
-c*d*(e-f)*(a*g-b*h)*(a*i-b*j)
-(a-b)*(c-d)*(a*d*f*g*i-b*c*e*h*j)
+(a-b)*(c*e-d*f)*(a*d*g*i-b*c*h*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 4 terms highest degree 5 of total 9
{ id10_4_5_9a = +a*b*g*h*i*j*(a-b)*(c-d)*(e-f)
-a*b*(g-h)*(i-j)*(a*d*f*g*i-b*c*e*h*j)
-(a-b)*(a*d*g*i-b*c*h*j)*(a*f*g*i-b*e*h*j)
+(a*g-b*h)*(a*i-b*j)*(a*d*f*g*i-b*c*e*h*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 4 terms highest degree 5 of total 9
{ id10_4_5_9b = +a*b*c*d*e*f*(e-f)*(g-h)*(i-j)
-e*f*(a-b)*(c-d)*(a*c*e*h*j-b*d*f*g*i)
-(e-f)*(a*c*e*h-b*d*f*g)*(a*c*e*j-b*d*f*i)
+(a*e-b*f)*(c*e-d*f)*(a*c*e*h*j-b*d*f*g*i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 5 terms highest degree 2 of total 4
{ id10_5_2_4a = +a*b*(i*f-g*j) -a*c*(e*i-g*h) +b*d*(e*j-f*h)
-(a*i-d*h)*(b*f-c*e) +(a*g-d*e)*(b*j-c*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 5 terms highest degree 2 of total 4
{ id10_5_2_4b = +a*b*c*e -a*b*d*f -(c*h-d*i)*(e*g-f*j)
+(c*j-d*g)*(e*i-f*h) -(c*e-d*f)*(a*b-g*h+i*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 5 terms highest degree 2 of total 4
{ id10_5_2_4c = +a*b*c*d -a*e*(d*j-g*i) -b*i*(c*h-e*f)
-(a*d-h*i)*(b*c-e*j) -e*i*(a*g+b*f-h*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 5 terms highest degree 2 of total 6
{ id10_5_2_6 = +a*c*f*i*(a-b)*(g-h) +a*b*f*(c-d)*(g-h)*(i-j)
+b*d*g*j*(a-b)*(e-f) -b*d*j*(a-b)*(e*g-f*h)
-f*(g-h)*(a*c-b*d)*(a*i-b*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Mark Haiman, Macdonald Polynomials and Geometry, pp. 207-254 in New
\\ Perspectives in Algebraic Combinatorics, 1999. (p. 214, equ. (2.14)) case
\\ n=3 : "\prod_{i=1}^n \frac{1-t^{-1}zx_i}{1-zx_i} = t^{-n}+\sum_{i=1}^n \frac
\\ {1}{1-zx_i}\frac{\prod_{j=1}^n (1-x_j/tx_i)}{\prod_{j\ne i} (1-x_j/x_i)}"
\\ {} 10 vars with 5 terms highest degree 3 of total 15
{ id10_5_3_15 =
+a*a*a*(c*e-d*f)*(c*g-d*h)*(c*i-d*j)*(e*h-f*g)*(e*j-f*i)*(g*j-h*i)
+(a-b)*d*f*(c*g-d*h)*(c*i-d*j)*(g*j-h*i)*(a*f*i-b*e*j)*(a*f*g-b*e*h)
-(a-b)*d*h*(c*e-d*f)*(c*i-d*j)*(e*j-f*i)*(a*e*h-b*f*g)*(a*h*i-b*g*j)
+(a-b)*d*j*(c*e-d*f)*(c*g-d*h)*(e*h-f*g)*(a*e*j-b*f*i)*(a*g*j-b*h*i)
-(e*h-f*g)*(e*j-f*i)*(g*j-h*i)*(a*c*e-b*d*f)*(a*c*g-b*d*h)*(a*c*i-b*d*j)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 6 terms highest degree 2 of total 5
{ id10_6_2_5a = +(a*c-b*d)*(e*g-f*h)*(i-j) -(a*c-b*d)*(e-f)*(g*i-h*j)
+(a*g-b*h)*(c*i-d*j)*(e-f) -(a*g-b*h)*(c*e-d*f)*(i-j)
+(a-b)*(c*e-d*f)*(g*i-h*j) -(a-b)*(c*i-d*j)*(e*g-f*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 6 terms highest degree 2 of total 5
{ id10_6_2_5b = +a*(c*e-d*f)*(g*i-h*j) -a*(c*i-d*j)*(e*g-f*h)
+e*(a*g-b*h)*(c*i-d*j) -e*(a*c-b*d)*(g*i-h*j)
+i*(a*c-b*d)*(e*g-f*h) -i*(a*g-b*h)*(c*e-d*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 6 terms highest degree 2 of total 5
{ id10_6_2_5c = +a*(b*c+e*j-f*i)*(b*g+d*e-f*h) +b*(a*f+c*d-g*j)*(a*e+c*h-g*i)
-c*(a*b+d*i-h*j)*(b*g+d*e-f*h) -d*(a*e+c*h-g*i)*(b*c+e*j-f*i)
-f*(a*b+d*i-h*j)*(a*e+c*h-g*i) +i*(a*f+c*d-g*j)*(b*g+d*e-f*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ A. Berkovich and K. Grizzell, "On the Class of Dominant and Subordinate
\\ Products. (p. 2) case L=5 : "$$ (1.6) \frac{1}{P(L)} -\frac{1}{Q(L)} =
\\ \sum_{i=1}^L \frac{Q(i-1)}{P(i)Q(L)} (\frac{Q(i)}{Q(i-1)} -\frac{P(i)}
\\ {P(i-1)}) $$
\\ {} 10 vars with 6 terms highest degree 2 of total 5
{ id10_6_2_5d = +a*c*e*g*(i-j) +a*c*e*(g*j-h*i) +a*c*i*(e*h-f*g)
+a*g*i*(c*f-d*e) +(a*d-b*c)*e*g*i -(a-b)*c*e*g*i ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into seven rectangles.
\\ {} 10 vars with 8 terms highest degree 1 of total 2 [NC]
{ id10_8_1_2 = +(a-b)*(h-j) +(a-c)*(f-h) -(a-e)*(f-j) +(b-c)*(h-i)
+(b-d)*(i-j) +(c-d)*(g-i) +(c-e)*(f-g) +(d-e)*(g-j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 10 vars with 10 terms highest degree 1 of total 2 [NC]
{ id10_10_1_2a = +(a-b)*(f+g) +(b-c)*(g+h) +(c-d)*(h+i) +(d-e)*(i+j)
-(a-e)*(f+j) -a*(g-j) +b*(f-h) +c*(g-i) +d*(h-j) -e*(f-i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Peter S. Bullen, A dictionary of inequalities, 1998, (p. 13) case n=5 :
\\ "Abel's Summation Formula, $$ \sum_{i=1}^n w_ia_i = W_n a_n - \sum_{i=1}
\\ ^{n-1} W_i\Delta a_i. (2) $$".
\\ Replace {w_1 -> a, w_2 -> c-a, w_3 -> e-c, w_4 -> g-e, w_5 -> -i, a_1 -> b,
\\ a_2 ->d, a_3 -> f, a_4 ->h, a_5 -> j}.
\\ {} 10 vars with 10 terms highest degree 1 of total 2 [NC]
{ id10_10_1_2b = -a*b +a*(b-d) +(a-c)*d +c*(d-f) +(c-e)*f +e*(f-h) +(e-g)*h
+g*(h-j) +(g-i)*j +i*j ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 11 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This identity is used in a general Somos-4/5 sequence proof :
\\ a2*(p0*a1*a6 +p2*a2*a5 -p3*a3*a4) +a1*(-p0*a2*a6 +p1*a3*a5 +p2*a4*a4) =
\\ a4*(p0*a0*a5 +p2*a1*a4 -p3*a2*a3) +a5*(-p0*a0*a4 +p1*a1*a3 +p2*a2*a2).
\\ Replace {a0 -> e, a1 -> a, a2 -> b, a3 -> f, a4 -> c, a5 -> d, a6 -> g,
\\ p0 -> k, p1 -> -h, p2 -> i, p3 -> j}.
\\ Replace {a -> a, b -> g*k, c -> b, d -> -c*i, e -> c, f -> d, g -> -f*h,
\\ h -> b*i, i -> e*k, j -> >f*j} in identity id10_4_2_3.
\\ {} 11 vars with 4 terms highest degree 3 of total 4
{ id11_4_3_4 = +a*(b*g*k-c*c*i+d*f*h) -b*(a*g*k+b*d*i-c*f*j)
+c*(a*c*i-b*f*j+d*e*k) -d*(a*f*h-b*b*i+c*e*k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This identity is used in a general Somos-4/5 sequence proof :
\\ a2*(a6*a0*p0^2-a4*a2*p2^2-a3^2*p3*p1) = a4*p2*(a4*a0*p0-a3*a1*p1-a2^2*p2)
\\ +a0*p0*(a6*a2*p0-a5*a3*p1-a4^2*p2) +a3*p1*(a5*a0*p0+a4*a1*p2-a3*a2*p3).
\\ Replace {a0 -> e, a1 -> b, a2 -> a, a3 -> f, a4 -> c, a5 -> d, a6 -> g,
\\ p0 -> k, p1 -> -h, p2 -> i, p3 -> j}.
\\ {} 11 vars with 4 terms highest degree 4 of total 5
{ id11_4_4_5a = +a*(a*c*i*i-f*f*h*j-e*g*k*k) -c*i*(a*a*i-b*f*h+c*e*k)
+e*k*(a*g*k+c*c*i-d*f*h) +f*h*(a*f*j-b*c*i+d*e*k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Replace {a -> a*i, b -> a, c -> f*j, d -> -c*i, e -> b, f -> e*k, g -> -d,
\\ h -> c, i -> -g*k, j -> f*h} in id10_4_2_3.
\\ {} 11 vars with 4 terms highest degree 4 of total 5
{ id11_4_4_5b = +a*i*(a*f*j-b*c*i+d*e*k) +b*(a*c*i*i-f*f*h*j-e*g*k*k)
-e*k*(a*d*i-b*g*k-c*f*j) -f*j*(a*a*i-b*f*h+c*e*k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 5 terms highest degree 2 of total 3
{ id11_5_2_3 = +a*(c*f+e*g+d*h) -b*(d*i-e*j) -d*(a*h-b*i) -c*(a*f-e*k)
-e*(a*g+b*j+c*k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 5 terms highest degree 2 of total 4
{ id11_5_2_4 = +(a*d+b*h+c*i)*(e*j+f*k) -(a*j+b*k)*(d*e+f*h+g*i)
-(a*f-b*e)*(d*k-h*j) +i*j*(a*g-c*e) +i*k*(b*g-c*f) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 5 terms highest degree 3 of total 15
{ id11_5_3_15 =
+b*f*i*(a*b-c*d)*(a*i-c*g)*(a*f-c*j)*(b*g-d*i)*(b*j-d*f)*(f*g-i*j)
-c*j*(e-f)*(a*b-c*d)*(a*i-c*g)*(b*g-d*i)*(f*g*h-i*i*j)*(b*b*j-d*f*k)
-c*g*(h-i)*(a*b-c*d)*(b*j-d*f)*(a*f-c*j)*(f*f*g-e*i*j)*(b*b*g-d*i*k)
+c*d*(b-k)*(a*f-c*j)*(a*i-c*g)*(f*g-i*j)*(b*e*j-d*f*f)*(b*g*h-d*i*i)
-(b*g-d*i)*(b*j-d*f)*(f*g-i*j)*(a*i*i-c*g*h)*(a*f*f-c*e*j)*(a*b*b-c*d*k)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 5 terms highest degree 4 of total 10
{ id11_5_4_10 = +d*f*j*(a-b)*(c*g-h*i)*(c*k-e*h)*(e*g-i*k)
+a*(c-d)*(e*g-i*k)*(c*f*k-e*e*h)*(c*g*j-h*i*i)
+a*(e-f)*(c*g-h*i)*(c*c*k-d*e*h)*(e*g*j-i*i*k)
+a*(i-j)*(c*k-e*h)*(c*c*g-d*h*i)*(e*e*g-f*i*k)
-(c*g-h*i)*(c*k-e*h)*(e*g-i*k)*(a*c*e*i-b*d*f*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 6 terms highest degree 2 of total 6
{ id11_6_2_6a = +(a*c-b*d)*(e*g-f*h)*(i*k-j*j)
-(a*c-b*d)*(g*i-h*j)*(e*k-f*j)
+(a*g-b*h)*(c*i-d*j)*(e*k-f*j)
-(c*e-d*f)*(a*g-b*h)*(i*k-j*j)
+(c*e-d*f)*(g*i-h*j)*(a*k-b*j)
-(e*g-f*h)*(c*i-d*j)*(a*k-b*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 6 terms highest degree 2 of total 6
{ id11_6_2_6b = +i*(a*c-b*d)*(e*g-f*h)*(j-k)
+(a*c-b*d)*(g*i-h*j)*(e*k-f*i)
-(a*g-b*h)*(c*i-d*j)*(e*k-f*i)
-i*(c*e-d*f)*(a*g-b*h)*(j-k)
-(c*e-d*f)*(g*i-h*j)*(a*k-b*i)
+(e*g-f*h)*(c*i-d*j)*(a*k-b*i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 6 terms highest degree 2 of total 6
{ id11_6_2_6c = +(a*c-b*d)*(e*g-f*h)*(h*j-i*k)
+(a*c-b*d)*(g*i-h*j)*(e*k-f*h)
-(a*g-b*h)*(c*i-d*j)*(e*k-f*h)
-(c*e-d*f)*(a*g-b*h)*(h*j-i*k)
-(c*e-d*f)*(g*i-h*j)*(a*k-b*h)
+(e*g-f*h)*(c*i-d*j)*(a*k-b*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 6 terms highest degree 3 of total 5
{ id11_6_3_5 = +a*b*b*(e*k-f*j-h*h) -a*b*d*(d*j-e*i-g*g)
-a*b*e*(b*k+d*i-e*h-f*g) +b*(b*f+d*d)*(a*j+c*h-d*g-e*f)
+(b*h-c*g-e*e)*(a*b*h+a*c*g-b*c*f-c*d*d)
-(a*g-b*f-d*d)*(b*d*g+b*e*f-c*c*g-c*e*e) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 7 terms highest degree 1 of total 5
{ id11_7_1_5 = +(a-k)*(b-k)*(c-k)*(d-k)*(e-k)
-(a-k)*(b-k)*(c-k)*(d-k)*(j-k)
-(a-k)*(b-k)*(c-k)*(i-k)*(e-j)
-(a-k)*(b-k)*(h-k)*(d-i)*(e-j)
-(a-k)*(g-k)*(c-h)*(d-i)*(e-j)
-(f-k)*(b-g)*(c-h)*(d-i)*(e-j)
-(a-f)*(b-g)*(c-h)*(d-i)*(e-j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into eight rectangles.
\\ Sherman K. Stein, Mathematics The Man-Made Universe, 2nd ed., W. H. Freeman,
\\ 1969, p. 116.
\\ {} 11 vars with 9 terms highest degree 1 of total 2 [NC]
{ id11_9_1_2a = +(a-b)*(g-j) +(a-d)*(j-k) -(a-f)*(g-k) +(b-c)*(h-j)
+(b-e)*(g-h) +(c-d)*(i-j) +(c-e)*(h-i) +(d-f)*(i-k)
+(e-f)*(g-i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into eight rectangles.
\\ Sherman K. Stein, Mathematics The Man-Made Universe, 2nd ed., W. H. Freeman,
\\ 1969, p. 121.
\\ {} 11 vars with 9 terms highest degree 1 of total 2 [NC]
{ id11_9_1_2b = +(a-b)*(h-k) +(a-c)*(g-h) -(a-f)*(g-k) +(b-c)*(h-i)
+(b-d)*(i-k) +(c-e)*(g-i) +(d-e)*(i-j) +(d-f)*(j-k)
+(e-f)*(g-j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 11 vars with 10 terms highest degree 1 of total 2
{ id11_10_1_2 = -(a-k)*(b-k) +(a-k)*(b-d) +(a-c)*(d-k) +(c-k)*(d-f)
+(c-e)*(f-k) +(e-k)*(f-h) +(e-g)*(h-k) +(g-k)*(h-j)
+(g-i)*(j-k) +(i-k)*(j-k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 12 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ The determinant | a b c d \\ e f g h \\ i j k l \\ a b c d| is 0.
\\ {} 12 vars with 4 terms highest degree 3 of total 4
{ id12_4_3_4 = +a*(-b*g*l+b*h*k+c*f*l-c*h*j-d*f*k+d*g*j)
+b*(a*g*l-a*h*k-c*e*l+c*h*i+d*e*k-d*g*i)
+c*(-a*f*l+a*h*j+b*e*l-b*h*i-d*e*j+d*f*i)
+d*(a*f*k-a*g*j-b*e*k+b*g*i+c*e*j-c*f*i) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "A Short Proof of an Identity of Sylvester", Internat. J.
\\ Math. & Math. Sci., v. 22, n. 2 (1999) 431-435, (p. 432, equ. 2.1) case n=3
\\ of an identity due to Milne: \sum_{r=1}^n (1-y_r) \prod_{i=1,i\ne r}^n
\\ \frac{1-x_iy_i/x_r}{1-x_i/x_r} = 1-y_1y_2\cdots y_n.
\\ {} 12 vars with 4 terms highest degree 3 of total 9
{ id12_4_3_9 = +(a*g-c*e)*(a*k-c*i)*(b*f*j-d*h*l)*(e*k-g*i)
-(j-l)*(a*g-c*e)*(a*b*k-c*d*i)*(e*f*k-g*h*i)
-(b-d)*(e*k-g*i)*(a*g*h-c*e*f)*(a*k*l-c*i*j)
-(f-h)*(a*k-c*i)*(a*b*g-c*d*e)*(e*k*l-g*i*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 12 vars with 4 terms highest degree 4 of total 7
{ id12_4_4_7 = +b*h*j*k*(c*f*h+d*e*h-d*f*g)
+f*(a*g-b*h)*(c*h*j*k-d*g*i*l-d*h*j*l)
+d*(g*i+h*j)*(a*f*g*l-b*e*h*k-b*f*h*l)
-g*h*k*(a*c*f*j-b*d*e*i-b*d*f*j) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Leonard E. Dickson, New First Course in the Theory of Equations, Wiley,
\\ 1939. (p. 140, Prob. 4) [a 2x2 det = sum of 3 products of 2x2 dets].
\\ From the Binet-Cauchy identity (AxB).(CxD) = (A.C)(B.D) - (B.C)(A.D)
\\ where A=(a,b,c), B=(e,f,g), C=(j,k,l), D=(d,h,i).
\\ {} 12 vars with 5 terms highest degree 2 of total 4
{ id12_5_2_4 = +(a*d+b*h+c*i)*(e*j+f*k+g*l)
-(a*j+b*k+c*l)*(d*e+f*h+g*i)
-(a*f-b*e)*(d*k-h*j)
-(a*g-c*e)*(d*l-i*j)
-(b*g-c*f)*(h*l-i*k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ Gaurav Bhatnagar, "A Short Proof of an Identity of Sylvester", Internat. J.
\\ Math. & Math. Sci., v. 22, n. 2 (1999) 431-435, (p. 432, equ. 2.1) case n=4
\\ of an identity due to Milne: \sum_{r=1}^n (1-y_r) \prod_{i=1,i\ne r}^n
\\ \frac{1-x_iy_i/x_r}{1-x_i/x_r} = 1-y_1y_2\cdots y_n.
\\ {} 12 vars with 5 terms highest degree 4 of total 16
{ id12_5_4_16 =
+(a*f-c*d)*(a*i-c*g)*(a*l-c*j)*(d*i-f*g)*(d*l-f*j)*(g*l-i*j)*(b*e*h*k-c*f*i*l)
-(b-c)*(d*i-f*g)*(d*l-f*j)*(g*l-i*j)*(a*f*f-c*d*e)*(a*i*i-c*g*h)*(a*l*l-c*j*k)
-(e-f)*(a*i-c*g)*(a*l-c*j)*(g*l-i*j)*(a*b*f-c*c*d)*(d*i*i-f*g*h)*(d*l*l-f*j*k)
+(h-i)*(a*f-c*d)*(a*l-c*j)*(d*l-f*j)*(a*b*i-c*c*g)*(d*e*i-f*f*g)*(i*j*k-g*l*l)
-(k-l)*(a*f-c*d)*(a*i-c*g)*(d*i-f*g)*(a*b*l-c*c*j)*(d*e*l-f*f*j)*(g*h*l-i*i*j)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ MR2178993 Zhang, Yusen and Chen, Wei
\\ q-triplicate inverse series relations with applications
\\ Rocky Mountain J. Math 35 (2005), 1407-1427. (p. 1417): "[[...]]
\\ We begin this section by giving a useful formula $$ (4.1) (1-d)(1-ea)(1-fa)
\\ +d(1-a)(1-b)(1-c) = (1-a)(1-bd)(1-cd) +a(1-d)(1-e)(1-f), $$ where efa=bcd."
\\ {} 12 vars with 5 terms highest degree 6 of total 8
{ id12_5_6_8 =
+(a-b)*(g-h)*(a*d*f*h*i*k-b*c*e*g*j*l) +a*b*d*f*h*(g-h)*(i-j)*(k-l)
-b*g*h*j*l*(a-b)*(c-d)*(e-f) +b*j*l*(a-b)*(c*g-d*h)*(e*g-f*h)
-d*f*h*(g-h)*(a*i-b*j)*(a*k-b*l)
; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is the determinant of a 3x3 matrix of rank 2.
\\ Shaun Cooper and Pee Choon Toh, "Determinant identities for theta
\\ functions", J. Math. Anal. Appl. 347 (2008), 1-7. (p. 3, Lemma 3.2):
\\ "$$ det(r_j s_k+t_j u_k)_{1\le j,k\le 3}=0. $$"
\\ {} 12 vars with 6 terms highest degree 2 of total 6
{ id12_6_2_6 = +(a*c-b*d)*(e*g-f*h)*(i*k-j*l)
-(a*c-b*d)*(e*k-f*l)*(g*i-h*j)
+(a*g-b*h)*(c*i-d*j)*(e*k-f*l)
-(a*g-b*h)*(c*e-d*f)*(i*k-j*l)
+(a*k-b*l)*(c*e-d*f)*(g*i-h*j)
-(a*k-b*l)*(c*i-d*j)*(e*g-f*h) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This identity is used to detect if a bilinear form in {v0,v1,v2}:
\\ v0*(b1*v1+b2*v2) + v1*(b3*v1+b4*v2) is equal to a product of two factors
\\ (a1*v1+a2*v2)*(a3*v0+a4*v1). This is, if 0=d*f-h*l=d*g-j*l=e*f-i*l=e*g-k*l,
\\ then (a*f+b*g)*(b*d+c*e) = l*( a*(b*h+c*i) +b*(b*j+c*k) ).
\\ {} 12 vars with 7 terms highest degree 2 of total 4
{ id12_7_2_4a = +a*b*(d*f-h*l) +a*c*(e*f-i*l) +b*b*(d*g-j*l) +b*c*(e*g-k*l)
-(a*f+b*g)*(b*d+c*e) +a*l*(b*h+c*i) +b*l*(b*j+c*k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This identity is used to detect if a bilinear form in {v0,v1,v2,v3}:
\\ v0*(b1*v1+b2*v2+b3*v3) +v1*(b4*v1+b5*v2+b6*v3) +v2*(b7*v2+b8*v3) is equal
\\ to a product of two factors like (a1*v1+a2*v2+a3*v3)*(a4*v0+a5*v1+a6*v2).
\\ That is, if 0=c*d-a*f=c*e-b*f-a*h=c*g-b*h, then (a*j+b*k+c*l)*(c*i+f*j+h*k)
\\ = c*( i*(a*j+b*k+c*l) +j*(d*j+e*k+f*l) +k*(g*k+h*l) ).
\\ {} 12 vars with 7 terms highest degree 2 of total 4
{ id12_7_2_4b = +(c*d-a*f)*j*j +(c*e-b*f-a*h)*j*k +(c*g-b*h)*k*k
-c*i*(a*j+b*k+c*l) -c*j*(d*j+e*k+f*l) -c*k*(g*k+h*l)
+(a*j+b*k+c*l)*(c*i+f*j+h*k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This identity is used to detect if a bilinear form in {v0,v1,v2,v3}:
\\ v0*(b1*v1+b2*v2+b3*v3) +v1*(b4*v1+b5*v2+b6*v3) +v2*(b7*v2+b8*v3) is equal
\\ to a product of two factors like (a1*v1+a2*v2)*(a4*v0+a5*v1+a6*v2+a7*v3).
\\ That is, if 0=c=a*h-b*f=a*a*g-a*b*e+b*b*d, then (a*b*i+b*d*j+a*g*k+b*f*l)*
\\ (a*j+b*k) = a*b*( i*(a*j+b*k+c*l) +j*(d*j+e*k+f*l) +k*(g*k+h*l) ).
\\ {} 12 vars with 7 terms highest degree 3 of total 5
{ id12_7_3_5 = +a*b*c*i*l +(a*b*e-a*a*g-b*b*d)*j*k +b*(a*h-b*f)*k*l
-a*b*i*(a*j+b*k+c*l) -a*b*j*(d*j+e*k+f*l) -a*b*k*(g*k+h*l)
+(a*j+b*k)*(a*b*i+b*d*j+a*g*k+b*f*l) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into nine rectangles.
\\ Sherman K. Stein, Mathematics The Man-Made Universe, 2nd ed., W. H. Freeman,
\\ 1969, p. 103, exer. 11.
\\ {} 12 vars with 10 terms highest degree 1 of total 2 [NC]
{ id12_10_1_2a = +(a-b)*(j-l) +(a-c)*(g-j) -(a-f)*(g-l) +(b-d)*(j-k)
+(b-f)*(k-l) +(c-d)*(i-j) +(c-e)*(h-i) +(c-f)*(g-h)
+(d-e)*(i-k) +(e-f)*(h-k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into nine rectangles.
\\ {} 12 vars with 10 terms highest degree 1 of total 2 [NC]
{ id12_10_1_2b = +(a-b)*(g-j) +(a-c)*(j-l) -(a-f)*(g-l) +(b-f)*(g-h)
+(b-e)*(h-i) +(b-d)*(i-j) +(c-f)*(k-l) +(c-d)*(j-k)
+(d-e)*(i-k) +(e-f)*(h-k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is a dissection of a rectangle into nine rectangles.
\\ Sherman K. Stein, Mathematics The Man-Made Universe, 2nd ed., W. H. Freeman,
\\ 1969, p. 98.
\\ {} 12 vars with 10 terms highest degree 1 of total 2 [NC]
{ id12_10_1_2c = +(a-b)*(g-j) +(a-c)*(j-l) -(a-f)*(g-l) +(b-d)*(g-i)
+(b-c)*(i-j) +(c-e)*(i-k) +(c-f)*(k-l) +(d-e)*(h-i)
+(d-f)*(g-h) +(e-f)*(h-k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 12 vars with 11 terms highest degree 1 of total 2 [NC]
{ id12_11_1_2a = +(a-f)*(a-g) +(a-b)*(g-i) +(b-e)*(g-h) +(e-f)*(g-k)
-(d-f)*(a-k) -(a-d)*(a-l) +(a-c)*(i-l) +(b-c)*(h-i)
+(c-e)*(h-j) +(d-e)*(j-k) +(c-d)*(j-l) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 12 vars with 11 terms highest degree 1 of total 2 [NC]
{ id12_11_1_2b = +(a-f)*(b-g) +(a-b)*(g-i) +(b-e)*(g-h) +(e-f)*(g-k)
-(d-f)*(b-k) -(a-d)*(b-l) +(a-c)*(i-l) +(b-c)*(h-i)
+(c-e)*(h-j) +(d-e)*(j-k) +(c-d)*(j-l) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 12 vars with 11 terms highest degree 1 of total 2 [NC]
{ id12_11_1_2c = +(a-f)*(c-g) +(a-b)*(g-i) +(b-e)*(g-h) +(e-f)*(g-k)
-(d-f)*(c-k) -(a-d)*(c-l) +(a-c)*(i-l) +(b-c)*(h-i)
+(c-e)*(h-j) +(d-e)*(j-k) +(c-d)*(j-l) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 12 vars with 12 terms highest degree 1 of total 2
{ id12_12_1_2 = +(a-b)*(g-h) -(b-c)*(h-i) +(c-d)*(i-j) -(d-e)*(j-k)
+(e-f)*(k-l) +(f-a)*(g-l) +a*(h-l) +b*(g-i) -c*(h-j)
+d*(i-k) -e*(j-l) -f*(g-k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ This is an identity for 2x2 matrices P,Q,R (see id6_7_1_2):
\\ $ det(P+Q+R) = det(P+Q)+det(Q+R)+det(P+R) -(det(P)+det(Q)+det(R)) $
\\ {} 12 vars with 14 terms highest degree 1 of total 2 [ZS]
{ id12_14_1_2 = +a*d -b*c +e*h -f*g +i*l -j*k -(a+e)*(d+h) +(b+f)*(c+g)
-(a+i)*(d+l) +(b+j)*(c+k) -(e+i)*(h+l) +(f+j)*(g+k)
+(a+e+i)*(d+h+l) -(b+f+j)*(c+g+k) ; }
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 13 variables \\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ {} 13 vars with 7 terms highest degree 2 of total 4
{ id13_7_2_4a = +a*b*(d*f-h*l) +a*c*(e*f-i*l) +b*b*(d*g-j*m) +b*c*(e*g-k*m)
-(a*f+b*g)*(b*d+c*e) +a*l*(b*h+c*i) +b*m*(b*j+c*k) ; }
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\\ {} 13 vars with 7 terms highest degree 2 of total 4
{ id13_7_2_4b = +(a*d-b*f)*k*k +(a*e-c*f-b*h)*k*l +(a*g-c*h)*l*l
-a*k*(d*k+e*l+f*m) -a*l*(g*l+h*m) -i*j*(a*m+b*k+c*l)
+(a*m+b*k+c*l)*(i*j+f*k+h*l) ; }
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\\ This is a dissection of a rectangle into ten rectangles.
\\ {} 13 vars with 11 terms highest degree 1 of total 2 [NC]
{ id13_11_1_2a = +(a-b)*(g-i) +(a-c)*(i-l) +(a-d)*(l-m) -(a-f)*(g-m)
+(b-e)*(g-h) +(b-c)*(h-i) +(e-f)*(g-k) +(d-f)*(k-m)
+(c-e)*(h-j) +(d-e)*(j-k) +(c-d)*(j-l) ; }
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\\ This is a dissection of a rectangle into ten rectangles.
\\ Sherman K. Stein, Mathematics The Man-Made Universe, 2nd ed., W. H. Freeman,
\\ 1969, p. 103, exer. 13.
\\ {} 13 vars with 11 terms highest degree 1 of total 2 [NC]
{ id13_11_1_2b = +(a-b)*(g-k) +(a-c)*(k-m) -(a-f)*(g-m) +(b-c)*(h-k)
+(b-d)*(g-h) +(c-d)*(h-j) +(c-e)*(j-l) +(c-f)*(l-m)
+(d-e)*(i-j) +(d-f)*(g-i) +(e-f)*(i-l) ; }
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\\ {} 13 vars with 12 terms highest degree 1 of total 2
{ id13_12_1_2 = +(a-b)*(g-h) -(b-c)*(h-i) +(c-d)*(i-j) -(d-e)*(j-k)
+(e-f)*(k-l) +(f-a)*(g-l) +(a-m)*(h-l) +(b-m)*(g-i)
-(c-m)*(h-j) +(d-m)*(i-k) -(e-m)*(j-l) -(f-m)*(g-k) ; }
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\\ 14 variables \\
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\\ {} 14 vars with 5 terms highest degree 3 of total 4
{ id14_5_3_4 = +a*(b*c*d+e*f*g+h*i*j) -d*(a*b*c+e*j*k+g*l*m)
-g*(b*j*n+a*e*f+d*h*m) +j*(a*i*l+b*g*n+d*e*k)
-(h+l)*(a*i*j-d*g*m) ; }
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\\ 15 variables \\
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\\ This is a dissection of a rectangle into thirteen rectangles.
\\ Sherman K. Stein, Mathematics The Man-Made Universe, 2nd ed., W. H. Freeman,
\\ 1969, p. 105, Exer. 30(a).
\\ {} 15 vars with 14 terms highest degree 1 of total 2 [NC]
{ id15_14_1_2 = -(a-g)*(h-o) +(a-e)*(h-i) +(e-g)*(h-j) +(a-b)*(i-l)
+(a-c)*(l-n) +(a-d)*(n-o) +(b-c)*(k-l) +(b-d)*(i-k)
+(c-f)*(k-m) +(c-d)*(m-n) +(d-e)*(i-j) +(d-g)*(m-o)
+(d-f)*(j-k) +(f-g)*(j-m) ; }
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\\ 18 variables \\
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\\ A. Cayley, C.M.P. v. 13, n. 911 "On an Algebraical Identity Relating to the
\\ Six Coordinates of a Line" (Mess. of Math., v 20 (1891), 138-140)
\\ {} 18 vars with 7 terms highest degree 3 of total 6
{ id18_7_3_6 =
+(-c*e*g+b*f*g+c*d*h-a*f*h-b*d*i+a*e*i)*(-f*h*j+e*i*j+f*g*k-d*i*k-e*g*l+d*h*l)
+(f*h-e*i)*(-f*h+e*i)*(a*j+d*m+g*p)
+(e*g-d*h)*(-f*h+e*i)*(c*j+a*l+f*m+d*o+i*p+g*r)
+(f*g-d*i)*(f*h-e*i)*(b*j+a*k+e*m+d*n+h*p+g*q)
+(e*g-d*h)*(-e*g+d*h)*(c*l+f*o+i*r)
+(e*g-d*h)*(f*g-d*i)*(c*k+b*l+f*n+e*o+i*q+h*r)
+(f*g-d*i)*(-f*g+d*i)*(b*k+e*n+h*q)
; }
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\\ This is a check if a single identity evaluates to zero.
{checkeqz(vx) = my(vy,vk,vt,vv,cnt=0);
if(type(vx)!=type([]), print("error: Invalid call"); return);
if(VERBOSE, print("Checking all identities now."));
for( n=1, length(vx), vy = vx[n]; vk = Vec(vy);
\\print(n": ",vy);
if( length(vk)<9, next); \\ minimum length "id0_0_0_0" is 9
if( vk[1]<>"i" || vk[2]<>"d" || vk[3]<"1" || vk[3]>"9", next);
vt = type(vv = eval(vy)); if( vt<>"t_POL", next); cnt++;
if(vv!=0, print("error ",vy," nonzero!!")));
if(VERBOSE, print("Check of ",cnt," identities done."));
if(idN!=cnt, print("error: Wrong number of identities!!"));
};
\\ check all available identities evaluate to zero. (using UNIX grep, cut)
checkeqz(externstr(Str("grep '^{ id' ",myfilename," | cut -d' ' -f2"))); \\}