Problem

Source: India IMOTC 2024 Day 3 Problem 2

Tags: number theory



Let $a$ and $n$ be positive integers such that: 1. $a^{2^n}-a$ is divisible by $n$, 2. $\sum\limits_{k=1}^{n} k^{2024}a^{2^k}$ is not divisible by $n$. Prove that $n$ has a prime factor smaller than $2024$. Proposed by Shantanu Nene