Glasnik Matematicki, Vol. 58, No. 1 (2023), 35-57. \( \)

ON THE \(D(4)\)-PAIRS \(\{a, ka\}\) WITH \(k\in \{2,3,6\}\)

Kouèssi Norbert Adédji, Marija Bliznac Trebješanin, Alan Filipin and Alain Togbé

Institut de Mathématiques et de Sciences Physiques, Université d'Abomey-Calavi, Bénin
e-mail:adedjnorb1988@gmail.com

Faculty of Science, University of Split, 21000 Split, Croatia
e-mail:marbli@pmfst.hr

Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia
e-mail:filipin@grad.hr

Department of Mathematics and Statistics, Purdue University Northwest, 1401 S, U.S. 421, Westville IN 46391, USA
e-mail:atogbe@pnw.edu

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 \[ d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))). \] As a corollary, we prove that any \(D(4)\)-quadruple tht contains the pair \(\{a, ka\}\) is regular.

2020 Mathematics Subject Classification.   11D09, 11B37, 11J68, 11J86

Key words and phrases.   Diophantine \(m\)-tuples, Pellian equations, Linear form in logarithms, Reduction method

Full text (PDF) (access from subscribing institutions only)

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

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