2021 USAMO Problems/Problem 3: Difference between revisions

Revision as of 23:16, 4 March 2021

The 2021 USAMO has not been out yet. Please do not edit this page. If you are a site admin, please delete this page.

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-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
+The 2021 USAMO has not been out yet. Please do not edit this page. If you are a site admin, please delete this page.
-(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>.