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2021 USAMO Problems/Problem 3

Revision as of 03:14, 3 March 2021 by Y.grace.yu (talk | contribs) (→‎Made-up USAMO problem -- (3) is unsolved...)
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A perfect number is a positive integer that is equal to the sum of its proper divisors, such as 6, 28, 496, and 8128. Prove that (1) All even perfect numbers follow the format $\frac{1}{2}M(M+1)$, where $M$ is a Mersenne prime; (2) All $\frac{1}{2}M*(M+1)$, where $M$ is a Mersenne prime, are even perfect numbers; (3) There are no odd perfect numbers.

Note: a Mersenne prime is a prime in the form of $2^p-1$.

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