2024 USAJMO Problems/Problem 3

Problem

Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq1$ . Suppose that $p>2$ is prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$ .

Solution 1

Lemma $1$ :

Given a prime $p$ , a positive integer $k$ , and an even $m$ such that $p^k|a(m)$ , we must have that $p^{k+1}|a(m+2)$ .

Proof of Lemma $1$ :

$a(m+1)\equiv (a(m))^{m+1}-1\equiv -1 \mod p^{k+1}$

Therefore, $a(m+2)\equiv (a(m+1))^{m+2}-1\equiv (-1)^{m+2}-1\equiv 0 \mod p^{k+1}$

2024 USAJMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6
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2024 USAJMO Problems/Problem 3

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Problem

Solution 1

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