Perfect Numbers

Numbers equal to the sum of their proper divisors — 6 = 1+2+3, 28 = 1+2+4+7+14 — connected deeply to Mersenne primes

2000
M₅ = 31
n perfect ⟺ σ(n) = 2n ⟺ n = 2^{p-1}(2^p − 1), 2^p−1 prime (Euler)

Every even perfect number has the form 2^(p-1)(2^p−1) where 2^p−1 is a Mersenne prime (Euler's theorem). Known perfect numbers: 6, 28, 496, 8128, 33550336. Whether any odd perfect numbers exist is one of the oldest open problems in mathematics — none found below 10^{1500}.