the sum of ALL divisors (including n) equals twice the number
A perfect number equals the sum of all its proper divisors (every divisor except itself). 6 = 1+2+3. 28 = 1+2+4+7+14. They are extraordinarily rare: only 51 are known, all even, and they grow astronomically. Whether any odd perfect number exists remains one of the oldest open problems in mathematics.
The first four perfect numbers: divisor portraits
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.
A perfect number equals the sum of its proper divisors: 6 = 1+2+3, 28 = 1+2+4+7+14. Euclid showed 2^(p-1)*(2^p-1) is perfect whenever 2^p-1 is prime. Euler proved the converse: every even perfect number has this form. Whether any odd perfect number exists is one of the oldest unsolved problems; none has ever been found. Only 51 perfect numbers are known, all even, corresponding to the 51 known Mersenne primes.