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A note on the exponential Diophantine equation $(4m^2+1)^x+(5m^2-1)^y=(3m)^z$

Volume 139 / 2015

Jianping Wang, Tingting Wang, Wenpeng Zhang Colloquium Mathematicum 139 (2015), 121-126 MSC: 11D61. DOI: 10.4064/cm139-1-7

Abstract

Let $m$ be a positive integer. Using an upper bound for the solutions of generalized Ramanujan–Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if $3\nmid m$, then the equation $(4m^2+1)^x+(5m^2-1)^y=(3m)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.

Authors

  • Jianping WangCollege of Science
    Chang'an University
    Xi'an, Shaanxi, P.R. China
    e-mail
  • Tingting WangDepartment of Mathematics
    Northwest University
    Xi'an, Shaanxi, P.R. China
    e-mail
  • Wenpeng ZhangDepartment of Mathematics
    Northwest University
    Xi'an, Shaanxi, P.R. China
    e-mail

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