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Power tower sum

Source: India IMOTC 2024 Day 3 Problem 2

May 31, 2024
number theory

Problem Statement

Let aa and nn be positive integers such that: 1. a2n−aa^{2^n}-a is divisible by nn, 2. ∑k=1nk2024a2k\sum\limits_{k=1}^{n} k^{2024}a^{2^k} is not divisible by nn.
Prove that nn has a prime factor smaller than 20242024.
Proposed by Shantanu Nene
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