2019/05/22 by Nikola Adžaga, Adžaga, Nikola, Alan Filipin +3
Mathematics · #11D09 #11J68 #11J86 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1905.09332
openalex publication_date 2019/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Franušić's previous paper, 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, 16k3-4k\ 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 16k3-4k 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.