engleski

On the extensions of the Diophantine triples in Gaussian integers

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, 16k3−4k}; ; 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 16k3 −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.

Diophantine m-tuples, Diophantine approximation, Pell equations

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