Question 96
Let $a_0, a_1, \dots$ be a sequence of positive integers such that $a_n = a_{n-1}^{2^n}$ for all $n = 1, 2, \dots$. Prove that for each prime $p$, $p \gt 3$, with residue $3$ modulo $4$ there exists a positive integer $a_0$ such that the sequence $a_N, a_{N+1}, a_{N+2}, \dots$ is not constant modulo $p$ for any positive integer $N$.