# Diophantine $m$-tuples and elliptic curves

• Volume: 13, Issue: 1, page 111-124
• ISSN: 1246-7405

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## Abstract

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A Diophantine $m$-tuple is a set of $m$ positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form ${y}^{2}=\left(ax+1\right)\left(bx+1\right)\left(cx+1\right)$, where $\left\{a,b,c\right\}$, is a Diophantine triple. In particular, we consider the elliptic curve ${E}_{k}$ defined by the equation ${y}^{2}=\left({F}_{2k}x+1\right)\left({F}_{2k+2}x+1\right)\left({F}_{2k+4}x+1\right),$ where $k\ge 2$ and ${F}_{n}$, denotes the $n$-th Fibonacci number. We prove that if the rank of ${E}_{k}\left(𝐐\right)$ is equal to one, or $k\le 50$, then all integer points on ${E}_{k}$ are given by$\begin{array}{c}\left(x,y\right)\in \left\{\left(0±1\right),\left(4{F}_{2k+1}{F}_{2k+2}{F}_{2k+3}±\left(2{F}_{2k+1}{F}_{2k+2}-1\right)\hfill \\ \hfill ×\left(2{F}_{2k+2}^{2}+1\right)\left(2{F}_{2k+2}{F}_{2k+3}+1\right)\right\}.\end{array}$

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