Complete solution of the polynomial version of a problem of Diophantus

Abstract

In this paper, we prove that if { a, b, c, d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then (a+b−c−d) 2=4(ab+1)(cd+1). This settles the “strong” Diophantine quintuple conjecture for polynomials with integer coefficients.