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Upper bound on integers satisfying n|3^n-1

Source: VJIMC 2024, Category I, Problem 4

April 14, 2024
number theoryAnalytic Number TheoryDivisibility

Problem Statement

Let p>2p>2 be a prime and let A={n∈N:2p∣n and p2∤n and n∣3n−1}.\mathcal{A}=\{n \in \mathbb{N}: 2p \mid n \text{ and } p^2\nmid n \text{ and } n \mid 3^n-1\}. Prove that lim sup⁡k→∞∣A∩[1,k]∣k≤2log⁡3plog⁡p.\limsup_{k \to \infty} \frac{\vert \mathcal{A} \cap [1,k]\vert}{k} \le \frac{2\log 3}{p\log p}.
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