On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$
Volume 161 / 2020
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.
