On the extensions of the Diophantine triples in Gaussian integers

Abstract

A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple {k−1, k+1, 16k 3 −4k} in Gaussian integers Z[i] to a Diophantine quadruple. Similar one-parameter family, {k − 1, k + 1, 4k}, was studied in [9],where it was shown that the extension to a Diophantine quadruple is unique (with an element 16k 3 − 4k). The family of the triples of the same form {k − 1, k + 1, 16k 3 − 4k} was studied in rational integers in [6]. It appeared as a special case while solving the extensibility problem of Diophantine pair {k − 1, k + 1}, in which it was not possible to use the same method as in the other cases. As authors (Bugeaud, Dujella and Mignotte) point out, the difficulty appears because the gap between k + 1 and 16k 3 − 4k is not sufficiently large. We find the same difficulty here while trying to use Diophantine approximations. Then we partially solve this problem by using linear forms in logarithms.