The diophantine equation $x^2+2^a\cdot {17}^b=y^n$

Gou, Su ; Wang, Tingting

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 645-654.

Résumé

Let $\mathbb{Z}$, $\mathbb{N}$ be the sets of all integers and positive integers, respectively. Let $p$ be a fixed odd prime. Recently, there have been many papers concerned with solutions $(x,y,n,a,b)$ of the equation $x^2+2^a{p}^b=y^n$, $x,y,n\in \mathbb{N}$, $gcd(x,y)=1$, $n\ge 3$, $a,b\in \mathbb{Z}$, $a\ge 0$, $b\ge 0.$ And all solutions of it have been determined for the cases $p=3$, $p=5$, $p=11$ and $p=13$. In this paper, we mainly concentrate on the case $p=3$, and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions $(x,y,n,a,b)$ of the equation $x^2+2^a\cdot {17}^b=y^n$, $x,y,n\in \mathbb{N}$, $gcd(x,y)=1$, $n\ge 3$, $a,b\in \mathbb{Z}$, $a\ge 0$, $b\ge 0$, are determined.

Détail

DOI : 10.1007/s10587-012-0056-z

Classification : 11D61
Mots-clés : exponential diophantine equation; modular approach; arithmetic properties of Lucas numbers