On the exponential Diophantine equation (18m(2 )+1)(x) + (7m(2)-1)(y) = (5m)(z)


ALAN M.

TURKISH JOURNAL OF MATHEMATICS, vol.42, no.4, pp.1990-1999, 2018 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 42 Issue: 4
  • Publication Date: 2018
  • Doi Number: 10.3906/mat-1801-76
  • Title of Journal : TURKISH JOURNAL OF MATHEMATICS
  • Page Numbers: pp.1990-1999

Abstract

Let m be a positive integer. We show that the exponential Diophantine equation (18m(2) + 1)(x) + (7m(2) - 1)(y) = (5m)(z) has only the positive integer solution (x, y, z) = (1, 1, 2) except for m 23, 47, 63, 87 (mod 120) . For m not equivalent to 0 (mod 5) we use some elementary methods and linear forms in two logarithms. For m 0 (mod 5) we apply a result for linear forms in p-adic logarithms.