Clemens Fuchs, TU Graz (Koautor: Andrej Dujella)
Let be an integer. A classical Diophantine -tuple with the proberty is a set of positive integers such that the product of any two of them increased by is a perfect square. There has been a lot of recent activity in this subject area, most notably by the coauthor, and the main problem of interest here is finding upper bounds on for which there can exist Diophantine -tuples satisfying . In this paper, we study this question in the ring , and we prove that for there is no Diophantine quadruple of non-zero polynomials, not all four constant, having the proberty . The main idea behind the proof is to reduce the given problem, via a theory of Pell equations in , to a question about occurrences of common values in two binary recurrent sequences of polynomials, a problem which is dealt with by looking at the degrees and leading terms of the polynomials arising as members of these two recurrent sequences.