Original scientific paper

https://doi.org/10.3336/gm.58.1.03

On the \(D(4)\)-pairs \(\{a, ka\}\) with \(k\in \{2,3,6\}\)

Kouèssi Norbert Adédji orcid.org/0000-0001-8257-5729 ; Institut de Mathématiques et de Sciences Physiques, Université d'Abomey-Calavi, Bénin
Marija Bliznac Trebješanin orcid.org/0000-0003-0640-5407 ; Faculty of Science, University of Split, 21000 Split, Croatia
Alan Filipin ; Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia
Alain Togbé orcid.org/0000-0002-5882-936X ; Department of Mathematics and Statistics, Purdue University Northwest, 1401 S, U.S. 421, Westville IN 46391, USA

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Abstract

Let \(a\) and \(b=ka\) be positive integers with \(k\in \{2, 3, 6\},\) such that \(ab+4\) is a perfect square. In this paper, we study the extensibility of the \(D(4)\)-pairs \(\{a, ka\}.\) More precisely, we prove that by considering families of positive integers \(c\) depending on \(a,\) if \(\{a, b, c, d\}\) is a set of positive integers which has the property that the product of any two of its elements increased by \(4\) is a perfect square, then \(d\) is given by
-17ex d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))).
As a corollary, we prove that any \(D(4)\)-quadruple tht contains the pair \(\{a, ka\}\) is regular.

Keywords

Diophantine \(m\)-tuples, Pellian equations, Linear form in logarithms, Reduction method

Hrčak ID:

304389

URI

https://hrcak.srce.hr/304389

Publication date:

20.6.2023.

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