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

Abstract

Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1{)}^x+(n+2{)}^y=n^z,n\ge 2,xyz\ne 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2),(3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.

12 pages, to appear, Colloquium Mathematicum (2020)