Problem

Source: Saudi Arabia IMO TST Day III Problem 1

Tags: number theory unsolved, number theory



A perfect number is an integer that equals half the sum of its positive divisors. For example, because $2 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28$, $28$ is a perfect number. (a) A square-free integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers. (b) Prove that no perfect square is a perfect number.