2021 USAMO Problems/Problem 3

A perfect number is a positive integer that is equal to the sum of its proper divisors, such as $6$ , $28$ , $496$ , and $8,128$ . 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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