Integer points on a family of elliptic curves

Abstract

Set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. First example of a Diophantine quadruple is found by Fermat, and it was {1, 3, 8, 120} (see [6, p. 517]). In 1969, Baker and Davenport [2] proved that if d is a positive integer such that {1, 3, 8, d} is a Diophantine quadruple, then d has to be 120. Recently, in [9], we generalized this result to all Diophantine triples of the form {1, 3, c}. The fact that {1, 3, c} is a Diophantine triple implies that c = ck for some positive integer k, where the sequence (ck) is given by c0 = 0, c1 = 8, ck+2 = 14ck+1 − ck + 8, k ≥ 0. Let ck + 1 = sk, 3ck + 1 = t 2 k. It is easy to check that ck±1ck + 1 = (2ck ± sktk). The main result of [9] is the following theorem.