On the exponential Diophantine equation (m2+1)x+(cm2−1)y=(am)z

Authors

  • T. Miyazaki College of Science and Technology Nihon University
  • N. Terai Ashikaga Institute of Technology

Keywords:

exponential Diophantine equation, integer solution, ower bound for linear forms in two logarithms

Abstract

Let m,a,c be positive integers with a≡3,5 (mod8). We show that when 1+c=a^2, the exponential Diophantine equation (m^2+1)^x+(cm^2−1)^y=(am)^z has only the positive integer solution (x,y,z)=(1,1,2) under the condition m≡±1 (moda), except for the case (m,a,c)=(1,3,8), where there are only two solutions: (x,y,z)=(1,1,2), (5,2,4). In particular, when a=3, the equation (m^2+1)^x+(8m^2−1)^y=(3m)^z has only the positive integer solution (x,y,z)=(1,1,2), except if m=1. The proof is based on elementary methods and Baker’s method. DOI: 10.1017/S0004972713000956

Published

2014-06-03

Issue

Vol. 90 No. 1 (2014)

Section

Articles
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