Naslov
On the extensions of the Diophantine triples in Gaussian integers

Autori
Adžaga, Nikola ; Filipin, Alan ; Franušić, Zrinka

Izvornik
Monatshefte für Mathematik (0026-9255) 197 (2022), 4; 535-563

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Diophantine m-tuples, Diophantine approximation, Pell equations

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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