Abstract
Let m>=2 and k>=2 be integers and let R be a commutative ring with a unit element denoted by 1. A k-th power diophantine m-tuple in R is an m-tuple (a_1, a_2, ..., a_m) of non-zero elements of R such that a_ia_j+1 is a k-th power of an element of R for 1 =3 and R=K[X], the ring of polynomials with coefficients in a field K of characteristic zero. We prove the following upper bounds on m, the size of diophantine m-tuple: m =5 ; m =8.
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