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.
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.
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