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

Luca F., Soydan G.

JOURNAL OF NUMBER THEORY, cilt.132, sa.11, ss.2604-2609, 2012 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 132 Sayı: 11
  • Basım Tarihi: 2012
  • Doi Numarası: 10.1016/j.jnt.2012.05.010
  • Dergi Adı: JOURNAL OF NUMBER THEORY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.2604-2609
  • Anahtar Kelimeler: Exponential Diophantine equations, Primitive divisors of Lehmer sequences
  • Bursa Uludağ Üniversitesi Adresli: Hayır

Özet

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.