On the extendibility of $D(4)$-pair of Pell numbers

Emanuel, D.; Kihel, O.; Moussaoui, K.; Zekiri, A.

doi:10.22124/jart.2025.29584.1764

On the extendibility of $D(4)$-pair of Pell numbers

Articles in Press

Document Type : Research Paper

Authors

¹ Department of Mathematics and Statistics‎, ‎Brock University‎, ‎L2S 3A1‎, ‎Saint Catharines Ontario‎, Canada

² Department of Algebra and Number Theory‎, ‎University of Sciences and Technology Houari Boumediene‎, Bab Ezzouar Algiers‎, ‎Algeria

³ Department of Algebra and Number Theory‎, ‎University of Sciences and Technology Houari Boumediene‎, ‎ Bab Ezzouar Algiers‎, ‎Algeria

10.22124/jart.2025.29584.1764

Abstract

Let $\ell$ be a non-zero integer, a set of $m$ distinct positive integers $\left\{a_{1},a_{2},\ldots,a_{m}\right\}$ is called a $D(\ell)$-Diophantine $m$-tuple, if $a_{i}a_{j}+\ell$ is a perfect square for any distinct $i,j\in\left\{1,2,...,m\right\}$. Let $P_n$ denote the $n^{th}$ Pell number, defined by the recurrence relation $P_{n+1}=2P_{n}+P_{n-1}$, with initial conditions $P_{0}=0$ and $P_{1}=1$. This paper investigates the extendibility of the pair $P_{2n+4}$, $4P_{2n+2}$ to a D(4)-Diophantine triple by another Pell number.

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