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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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Sažetak

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.

Ključne riječi

Diophantine $m$-tuples, Pellian equations, Linear form in logarithms, Reduction method

Hrčak ID:

304389

URI

https://hrcak.srce.hr/304389

Datum izdavanja:

20.6.2023.

Posjeta: 647 *