Find a number that exactly equals the sum of its own divisors
Six is divisible by 1, 2, and 3 — and 1+2+3 happens to equal 6 exactly. The next such number is 28 (1+2+4+7+14=28). These are called perfect numbers. Is there a way to generate every even one, and are there infinitely many?
Reveal the answer
Euclid showed over 2,000 years ago that whenever 2^p − 1 is prime (a "Mersenne prime"), the number 2^(p−1) × (2^p − 1) is perfect; two thousand years later, Euler proved every even perfect number must have this exact form. Whether infinitely many exist is still unknown, since it depends on whether infinitely many Mersenne primes exist — and whether any odd perfect number exists at all remains one of the oldest unsolved problems in mathematics.