The Refined Robinson Polynomial Sequence Conjecture

                       by Michael Somos 11 Apr 1999

Given integers { k , p , q , r } such that 6 <= k , 0 < p < q < r <= k/2 .

Given variables { x , y , z } , define a sequence by

    a(1) = a(2) = ... = a(k) = 1 , and

           x * a(n-p)*a(n-k+p) + y * a(n-q)*a(n-k+q) + z * a(n-r)*a(n-k+r)
    a(n) = --------------------------------------------------------------- .
                                        a(n-k)

By definition, a(n) is a rational function in variables { x , y , z } with
non-negative integer coefficients.

Conjecture: 
      a(n) is always a polynomial if and only if p+q = r or k-r .
      It first fails to be a polynomial at a(n) when the denominator
      a(n-k) = x+y+z or x(x+y+z)+y+z , and when k+1 <= n-k <= k+1+p .
end.

References:

Richard K. Guy, Unsolved Problems in Number Theory, 2nd edition, page 215.
"E15 ... Raphael Robinson has observed ... x_nx_{n-k}=ax_{n-p}x_{n-k+p}+
bx_{n-q}x_{n-k+q}+cx_{n-r}x_{n-k+r}  ... appears to generate integers from
the starting values ... a>=0,b>=0,c>=0, p>=1,q>=1,r>=1, p+q+r=k."

David Gale, The Mathematical Intelligencer, Vol 13, No. 1, 1991, page 42.
"... Conjecture: For any p, q, r < k the recursion a_n a_{n-k} =
      xa_{n-p}a_{n-k+p} + ya_{n-q}a_{n-k+q} + za_{n-r}a_{n-k+r}       (7)
generates integers if and only if p, q, r can be chosen so that p+q+r=k.
(Robinson's evidence is only for the case x = y = z = 1. The arbitrary
x, y, z are my responsibility.)"


Further Conjecture by Michael Somos 12 Apr 1999 : Given integers { k, p, q, r, s } such that 8 <= k , 0 < p < q < r < s <= k/2 . Given variables { x, y, z, w } , define a sequence by a(1) = a(2) = ... = a(k) = 1 , and x*a(n-p)*a(n-k+p)+y*a(n-q)*a(n-k+q)+z*a(n-r)*a(n-k+r)+w*a(n-s)*a(n-k+s) a(n)=-----------------------------------------------------------------------. a(n-k) Conjecture: a(n) always fails to be a polynomial at a(n) when the denominator a(n-k) = x+y+z+w or x^2+(y+z+w)*x+y+z+w and when k+1 <= n-k <= k+1+p . end. Comment: The Somos-4, Somos-5, Somos-6, Somos-7 sequences are all special cases of the Robinson sequence conjecture. After that, his sequences produce integers and mine don't.

Back to my math page
Back to my home page
Last Updated Mon Aug 9 01:50 EDT 1999
Michael Somos <somos@grail.cba.csuohio.edu>
WWW URL: "http://grail.cba.csuohio.edu/~somos/"