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arXiv · 2206.12842

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

Kouèssi Norbert Adédji·Marija Bliznac Trebješanin·Alan Filipin·Alain Togbé
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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 three families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the 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 $d$ in given by $$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a, ka\}$ is regular.

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Kouèssi Norbert Adédji, Marija Bliznac Trebješanin, Alan Filipin, Alain Togbé. 2022-06-26. On the $D(4)$-pairs $\{a, ka\}$ with $k\in \{2,3,6\}$. https://doi.org/10.3336/gm.58.1.03

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