Difference between revisions of "2021 USAMO Problems/Problem 3"
Y.grace.yu (talk | contribs) (→Made-up USAMO problem -- (3) is unsolved...) |
(No difference)
|
Revision as of 04:14, 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 8128. 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 
.
