Problem

Source: Iranian third round 2018 number theory exam problem 2

Tags: number theory



Prove that for every prime number $p$ there exist infinity many natural numbers $n$ so that they satisfy: $2^{2^{2^{ \dots ^{2^n}}}} \equiv n^{2^{2^{\dots ^{2}}}} (mod p)$ Where in both sides $2$ appeared $1397$ times