The rule
perfect if the proper divisors add to the number · Euclid: 2^(p−1) × (2^p − 1)
Why it works
A perfect number equals the sum of its proper divisors. Euclid showed that whenever 2^p − 1 is prime, the product 2^(p−1) × (2^p − 1) is perfect, and Euler proved two thousand years later that every even perfect number has that form.
How to do it by hand
- For each number, find its proper divisors
- Add them up
- If the total equals the number, it is perfect
- Check it against Euclid's formula
What is worth knowing
Perfect numbers are astonishingly scarce: 6, 28, 496, 8128, and then nothing until 33,550,336. Only 51 are known, and each new one is found alongside a new Mersenne prime, because the two are tied by Euclid's formula. The primes 2^p − 1 that generate them are named after Mersenne, and searching for them is what the distributed GIMPS project has been doing since 1996. Two things remain unknown after two millennia: whether any odd perfect number exists, and whether the even ones ever run out.