Diophantine $m$-tuples and elliptic curves

Dujella, Andrej

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 111-124.

Un $m$-uplet diophantien est un ensemble de $m$ entiers naturels non nuls tel que le produit quelconque de deux d’entre eux augmenté de $1$ est un carré parfait. Dans cet article, nous nous intéressons a certaines propriétés de courbes elliptiques d’équation du type $y^2=(ax+1)(bx+1)(cx+1)$, où $\{a,b,c\}$ est un triplet diophantien. Nous considérons en particulier la courbe elliptique $E_k$ définie par l’équation $y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1),$ où $k\ge 2$ et $F_n$ désigne le $n$-ème nombre de Fibonacci. Nous montrons que si le rang de $E_k$ est égal a $1$, ou si $k\le 50$, alors les points entiers sur $E_k$ sont donnés par

$$\begin{array}{c}(x,y)\in \{(0\pm 1),(4F_{2k+1}{F}_{2k+2}{F}_{2k+3}\pm \left(2F_{2k+1}{F}_{2k+2}-1\right)\hfill \\ \hfill \times \left(2F_{2k+2}^2+1\right)\left(2F_{2k+2}{F}_{2k+3}+1\right)\}.\end{array}$$

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 $y^2=(ax+1)(bx+1)(cx+1)$, where $\{a,b,c\}$, is a Diophantine triple. In particular, we consider the elliptic curve $E_k$ defined by the equation $y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1),$ 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

$$\begin{array}{c}(x,y)\in \{(0\pm 1),(4F_{2k+1}{F}_{2k+2}{F}_{2k+3}\pm \left(2F_{2k+1}{F}_{2k+2}-1\right)\hfill \\ \hfill \times \left(2F_{2k+2}^2+1\right)\left(2F_{2k+2}{F}_{2k+3}+1\right)\}.\end{array}$$

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