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 fro, 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 (836-9ot): if p = 3.2m -1, q= 3.2m-1 -1 and r= 9.22m-1 — t, where m is an integer and p,q,r prime numbers, then 2m pq and 2m 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), Rene Descartes, to whom the formula of Tobit ben Korra is some-times ascribed, C . Rudolphus and others .