A pair of numbers with the property that each is equal to the sum of the proper positive divisors of the other. For example, 220 and 284 are amicable numbers because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, whose sum is 284, and the proper divisors of 284 are 1, 2, 4, 71, and 142, whose sum is 220.

These numbers, known to the Pythagoreans, were used as symbols of friendship. The amicable numbers 17 296 and 18 416 were found by Fermat, and a list of 64 pairs was produced by Euler. In 1867, a 16-year-old Italian boy found the second smallest pair, 1184 and 1210, overlooked by Euler. More than a billion pairs are now known. It is unknown whether or not there are infinitely many pairs of amicable numbers.