On the exponential Diophantine equation \((m^2+1)^x+(cm^2-1)^y=(am)^z \)

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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