On the Diophantine equation 2(m) + nx(2) = y(n)


Luca F., Soydan G.

JOURNAL OF NUMBER THEORY, vol.132, no.11, pp.2604-2609, 2012 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 132 Issue: 11
  • Publication Date: 2012
  • Doi Number: 10.1016/j.jnt.2012.05.010
  • Journal Name: JOURNAL OF NUMBER THEORY
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.2604-2609
  • Keywords: Exponential Diophantine equations, Primitive divisors of Lehmer sequences

Abstract

In this note, we prove that the Diophantine equation 2(m) + nx(2) = y(n) in positive integers x, y, m, n has the only solution (x, y,m,n) = (21,11,3,3) with n > 1 and gcd(nx, y) = 1. In fact, for n = 3,15, we transform the above equation into several elliptic curves for which we determine all their {2}-integer points. For n not equal 3,15, we apply the result of Yu.F. Bilu, G. Hanrot and P.M. Voutier about primitive divisors of Lehmer sequences. (C) 2012 Elsevier Inc. All rights reserved.