Question 36
Let $x_1$ be a fixed positive integer. Prove that there is no positive integer $m \gt 2500$ such that the sequence $x_{n+1} = x_n^{S(n)} + 1$ is eventually periodic modulo $m$, where $S(n)$ is the sum of digits of $n$ in base 10.
(We say a sequence is eventually periodic mod $m$ if there are positive integers $N, T$ such that $x_n \equiv x_{n+T} \pmod m$ for all $n \ge N$.)
(We say a sequence is eventually periodic mod $m$ if there are positive integers $N, T$ such that $x_n \equiv x_{n+T} \pmod m$ for all $n \ge N$.)