Diophantine m-tuples and elliptic curves
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),k2 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 k50, alors les points entiers sur E k sont donnés par

(x,y){(0±1),(4F 2k+1 F 2k+2 F 2k+3 ±2F 2k+1 F 2k+2 -1×2F 2k+2 2 +12F 2k+2 F 2k+3 +1}.

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 k2 and F n , denotes the n-th Fibonacci number. We prove that if the rank of E k (𝐐) is equal to one, or k50, then all integer points on E k are given by

(x,y){(0±1),(4F 2k+1 F 2k+2 F 2k+3 ±2F 2k+1 F 2k+2 -1×2F 2k+2 2 +12F 2k+2 F 2k+3 +1}.

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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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