Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples

Abstract

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by 1 is a perfect square. A conjecture in this field asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple in some sense. This problem is reduced to study the coincidence between certain two binary recurrent sequences of integers. As an analogy of this, we consider a similar coincidence on the polynomial ring in one variable over the integers, and determine it completely. Our result is regarded as a generalization of a result in the paper “Complete solution of the polynomial version of a problem of Diophantus” by A. Dujella, C. Fuchs in Journal of Number Theory 106 (2004) on the polynomial variant of Diophantine tuples.