Diophantine $m$-tuples and elliptic curves

Dujella, Andrej

Diophantine

m

-tuples and elliptic curves

Dujella, Andrej

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

Résumé
Abstract

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 $y^{2} = (a x + 1) (b x + 1) (c x + 1)$ , où ${a, b, c}$ est un triplet diophantien. Nous considérons en particulier la courbe elliptique $E_{k}$ définie par l’équation $y^{2} = (F_{2 k} x + 1) (F_{2 k + 2} x + 1) (F_{2 k + 4} x + 1),$ où $k \geq 2$ 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 $k \leq 50$ , alors les points entiers sur $E_{k}$ sont donnés par

\begin{matrix} (x, y) \in {(0 \pm 1), (4 F_{2 k + 1} F_{2 k + 2} F_{2 k + 3} \pm (2 F_{2 k + 1} F_{2 k + 2} - 1) \\ \times (2 F_{2 k + 2}^{2} + 1) (2 F_{2 k + 2} F_{2 k + 3} + 1)} . \end{matrix}

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 $y^{2} = (a x + 1) (b x + 1) (c x + 1)$ , where ${a, b, c}$ , is a Diophantine triple. In particular, we consider the elliptic curve $E_{k}$ defined by the equation $y^{2} = (F_{2 k} x + 1) (F_{2 k + 2} x + 1) (F_{2 k + 4} x + 1),$ where $k \geq 2$ and $F_{n}$ , denotes the $n$ -th Fibonacci number. We prove that if the rank of $E_{k} (𝐐)$ is equal to one, or $k \leq 50$ , then all integer points on $E_{k}$ are given by

\begin{matrix} (x, y) \in {(0 \pm 1), (4 F_{2 k + 1} F_{2 k + 2} F_{2 k + 3} \pm (2 F_{2 k + 1} F_{2 k + 2} - 1) \\ \times (2 F_{2 k + 2}^{2} + 1) (2 F_{2 k + 2} F_{2 k + 3} + 1)} . \end{matrix}

MR Zbl

@article{JTNB_2001__13_1_111_0,
     author = {Dujella, Andrej},
     title = {Diophantine $m$-tuples and elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {111--124},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     mrnumber = {1838074},
     zbl = {1046.11034},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_1_111_0/}
}

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Dujella, Andrej. Diophantine $m$-tuples and elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 111-124. http://www.numdam.org/item/JTNB_2001__13_1_111_0/

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