On the extensions of the Diophantine triples in Gaussian integers

Scientific paper - Original scientific paper

• Author: Nikola Adžaga
• Author: Alan Filipin
• Author: Zrinka Franušić
• Author's institution: University of Zagreb; Faculty of Civil Engineering
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Civil Engineering

Abstract (english)

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.

• Keywords (english): Diophantine m-tuples, Diophantine approximation, Pell equations
• Language: english
• Publication type: Scientific paper - Original scientific paper
• Publication status: Published
• Peer review: Peer review - international
• Publication version: Published version
• Journal title: Monatshefte für Mathematik
• Numbering: vol. 197(4), pp. 535-563
• e-ISSN: 1436-5081
• URN:NBN: urn:nbn:hr:237:781222
• Publication: 2022
• Type of resource: Text
• Access conditions: Open access
• Created on: 2024-09-05 05:10:53