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 ${\mathbb {Z}}{[i]}$ to a Diophantine quadruple. Similar one-parameter family, $\{k-1, k+1, 4k\}$, was studied in Franušić (Glasnik matematički 43(2):265–291, 2008), 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 Bugeaud et al. (Glasg Math J 49:333–344, 2007). 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.