Friendly Numbers

Pythagoras said "220 and 284 are friendly numbers" !! These numbers have a special property : Each is equal to the sum of the other's proper divisors. Proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 (they sum to 284). Proper divisors of 284 are 1, 2, 4, 71 and 142 (they sum to 220). More example include (1184, 1210), (17296, 18416). It is not known whether there are infinitely many friendly numbers. Twin primes are a pair of consecutive odd numbers both of which are prime.

• Theorem : There are infinitely many friendly numbers. (proof)

• Conjecture : There are infinitely many twin primes.