Diophantine m -tuples and elliptic curves

Andrej Dujella

Journal de théorie des nombres de Bordeaux (2001)

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

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 = ( 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 ≥ 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 ≤ 50 , then all integer points on E k are given by ( x , y ) ∈ { ( 0 ± 1 ) , ( 4 F 2 k + 1 F 2 k + 2 F 2 k + 3 ± 2 F 2 k + 1 F 2 k + 2 - 1 × 2 F 2 k + 2 2 + 1 2 F 2 k + 2 F 2 k + 3 + 1 } .

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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