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 come exactly from Mersenne primes
Perfect numbers on a log scale: they grow faster than exponentially
Even on a log scale, where each unit of screen space represents a factor of 10 - the jump from 8,128 to 33,550,336 spans most of the chart. The 51st known perfect number, found in 2024, has over 49 million digits and would fill a small book.
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.