1911 Encyclopædia Britannica/Amicable Numbers

AMICABLE NUMBERS, two numbers so related that the sum of the factors of the one is equal to the other, unity being considered as a factor. Such a pair are 220 and 284; for the factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the factors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220. Amicable numbers were known to the Pythagoreans, who accredited them with many mystical properties. A general formula by which these numbers could be derived was invented by the Arabian astronomer Tobit ben Korra (836–901): if p = 3·2^m − 1, q = 3·2^m−1 − 1 and r = 9·2^2m−1 − 1, where m is an integer and p, q, r prime numbers, then 2^m pq and 2^m r are a pair of amicable numbers. This formula gives the pairs 220 and 284, 17,296 and 18,416, 9,463,584 and 9,437,056. The pair 6232 and 6368 are amicable, but they cannot be derived from this formula. Amicable numbers have been studied by Al Madshritti (d. 1007), René Descartes, to whom the formula of Tobit ben Korra is sometimes ascribed, C. Rudolphus and others.