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 ${k-1, k+1, 16k<sup>3-4k}$ in Gaussian integers $\mathbb{Z}[i]$ to a Diophantine quadruple. Similar one-parameter family, ${k-1, k+1, 4k}$, was studied in Franu\v{s}i\'c's previous paper, 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 ${k-1, k+1, 16k<sup>3-4k}$ was already studied in rational integers. 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.