Problem

Source: Iran MO Third Round 2021 N3

Tags: number theory



$x_1$ is a natural constant. Prove that there does not exist any natural number $m> 2500$ such that the recursive sequence $\{x_i\} _{i=1} ^ \infty $ defined by $x_{n+1} = x_n^{s(n)} + 1$ becomes eventually periodic modulo $m$. (That is there does not exist natural numbers $N$ and $T$ such that for each $n\geq N$, $m\mid x_n - x_{n+T}$). ($s(n)$ is the sum of digits of $n$.)