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.
"Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples." Tokyo J. Math. 42 (2) 611 - 619, December 2019. https://doi.org/10.3836/tjm/1502179292