Euclid–Euler theorem: even perfect numbers ↔ Mersenne primes
n is even perfect ⟺ n = 2^(p−1) · (2^p − 1)
where 2^p − 1 is a Mersenne prime
Euclid proved the → direction. Euler proved ← . All 51 known perfect numbers are even and come from this formula. Whether odd perfect numbers exist is unknown.
Perfect numbers on a log scale: they grow faster than exponentially
Values shown as log10. Even on a log scale each jump is dramatically larger. The 51st perfect number has over 49 million digits.