Difference between revisions of "2021 USAMO Problems/Problem 3"

Revision as of 03:15, 3 March 2021

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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-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
+A perfect number is a positive integer that is equal to the sum of its proper divisors, such as <math>6</math>, <math>28</math>, <math>496</math>, and <math>8,128</math>. Prove that
 (1) All even perfect numbers follow the format <math>\frac{1}{2}M(M+1)</math>, where <math>M</math> is a Mersenne prime;
 (2) All <math>\frac{1}{2}M*(M+1)</math>, where <math>M</math> 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 <math>2^p-1</math>.