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

## How to cite

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Dujella, Andrej. "Diophantine $m$-tuples and elliptic curves." Journal de théorie des nombres de Bordeaux 13.1 (2001): 111-124. <http://eudml.org/doc/248693>.

@article{Dujella2001,
abstract = {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 $\lbrace a, b, c\rbrace$, 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 $E_k (\mathbf \{Q\})$ is equal to one, or $k \le 50$, then all integer points on $E_k$ are given by\begin\{multline*\}(x, y) \in \lbrace (0 \pm 1), (4F\_\{2k+1\} F\_\{2k+2\}F\_\{2k+3\} \pm \left( 2F\_\{2k+1\} F\_\{2k+2\} - 1 \right)\\ \times \left( 2 F^2\_\{2k+2\} + 1 \right) \left( 2 F\_\{2k+2\} F\_\{2k+3\} + 1 \right) \rbrace .\end\{multline*\}},
author = {Dujella, Andrej},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {111-124},
publisher = {Université Bordeaux I},
title = {Diophantine $m$-tuples and elliptic curves},
url = {http://eudml.org/doc/248693},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Dujella, Andrej
TI - Diophantine $m$-tuples and elliptic curves
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 111
EP - 124
AB - 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 $\lbrace a, b, c\rbrace$, 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 $E_k (\mathbf {Q})$ is equal to one, or $k \le 50$, then all integer points on $E_k$ are given by\begin{multline*}(x, y) \in \lbrace (0 \pm 1), (4F_{2k+1} F_{2k+2}F_{2k+3} \pm \left( 2F_{2k+1} F_{2k+2} - 1 \right)\\ \times \left( 2 F^2_{2k+2} + 1 \right) \left( 2 F_{2k+2} F_{2k+3} + 1 \right) \rbrace .\end{multline*}
LA - eng
UR - http://eudml.org/doc/248693
ER -

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