A pair of numbers with the property that each is equal to the sum of the positive divisors of the other. (For the purposes of this definition, a number is not included as one of its own divisors.) For example, 220 and 284 are amicable numbers because the positive divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, whose sum is 284, and the positive 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 sixteen-year-old Italian boy found the second smallest pair, 1184 and 1210, overlooked by Euler. More than 600 pairs are now known. It has not been shown whether or not there are infinitely many pairs of amicable numbers.