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On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$

Volume 161 / 2020

Hairong Bai, Elif Kızıldere, Gökhan Soydan, Pingzhi Yuan Colloquium Mathematicum 161 (2020), 239-249 MSC: Primary 11D61; Secondary 11D41. DOI: 10.4064/cm7668-6-2019 Published online: 30 March 2020

Abstract

Suppose that $n$ is a positive integer. We show that the only positive integer solutions $(n,x,y,z)$ of the exponential Diophantine equation $$ (n-1)^{x}+(n+2)^{y}=n^{z}, \quad n\geq 2, \, xyz\neq 0, $$ are $(3,2,1,2), (3,1,2,3)$. The main tools in the proofs are Baker’s theory and Bilu–Hanrot–Voutier’s result on primitive divisors of Lucas numbers.

Authors

  • Hairong BaiSchool of Mathematics
    South China Normal University
    Guangzhou 510631, China
    e-mail
  • Elif KızıldereDepartment of Mathematics
    Bursa Uludağ University
    Bursa 16059, Turkey
    e-mail
  • Gökhan SoydanDepartment of Mathematics
    Bursa Uludağ University
    Bursa 16059, Turkey
    e-mail
  • Pingzhi YuanSchool of Mathematics
    South China Normal University
    Guangzhou 510631, China
    e-mail

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