Difference between revisions of "2024 USAJMO Problems/Problem 3"

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

Revision as of 20:44, 19 March 2024

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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 $p^k$.