Publication date: 30 June 2023
Volume: Vol 58
Issue: Svezak 1
Pages: 35-57
DOI: 10.3336/gm.58.1.03
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
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.
Diophantine \(m\)-tuples, Pellian equations, Linear form in logarithms, Reduction method
304389
20.6.2023.
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