A Diophantine -tuple is a set of 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 , where , is a Diophantine triple. In particular, we consider the elliptic curve defined by the equation where and , denotes the -th Fibonacci number. We prove that if the rank of is equal to one, or , then all integer points on are given by
Un -uplet diophantien est un ensemble de entiers naturels non nuls tel que le produit quelconque de deux d’entre eux augmenté de est un carré parfait. Dans cet article, nous nous intéressons a certaines propriétés de courbes elliptiques d’équation du type , où est un triplet diophantien. Nous considérons en particulier la courbe elliptique définie par l’équation où et désigne le -ème nombre de Fibonacci. Nous montrons que si le rang de est égal a , ou si , alors les points entiers sur sont donnés par
@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://archive.numdam.org/item/JTNB_2001__13_1_111_0/} }
Dujella, Andrej. Diophantine $m$-tuples and elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 111-124. http://archive.numdam.org/item/JTNB_2001__13_1_111_0/
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