Problem

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

Tags: number theory



$a,b,m,k\in \mathbb{Z}$, $a,b,m>2,k>1$, for which $k^n a+b$ is an $m$-th power of a natural number for $\forall n\in \mathbb{N}$. Prove that $b$ is an $m$-th power of a non-negative integer.