2021 USAMO Problems/Problem 3: Difference between revisions
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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 | 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; | (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; | (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. | (3) There are no odd perfect numbers. | ||
Note: a Mersenne prime is a prime in the form of <math>2^p-1</math>. | Note: a Mersenne prime is a prime in the form of <math>2^p-1</math>. | ||
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 
, 
, 
, and 
. Prove that
(1) All even perfect numbers follow the format 
, where 
is a Mersenne prime;
(2) All 
, where 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 
.
