| Amicable Number |
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| number theory | |
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A pair of amicable numbers constitutes an Aliquot Sequence of Period 2 . A general formula by which these numbers could be derived was invented circa 850 by Thabit Ibn Qurra ( 826 - 901 ): if p q r where ''n'' > 1 is an Integer and ''p'', ''q'', and ''r'' are Prime Number s, then 2''npq'' and 2''nr'' are a pair of amicable numbers. This formula gives the amicable pair (220, 284), as well as the pair (17,296, 18,416) and the pair (9,363,584, 9,437,056). The pair (6232, 6368) are amicable, but they cannot be derived from this formula. In fact, this formula produces amicable numbers for ''n'' = 2, 4, and 7, but for no other values below 20,000. In every known case, the numbers of a pair are either both Even or both Odd , though there is no known reason why an even-odd pair could not exist. Also, every known pair shares at least one common Factor . It is not known whether a pair of Coprime amicable numbers exist, though if they do, their Product must be greater than 1067. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula. Amicable numbers have been studied by Al Madshritti (died 1007 ), Abu Mansur Tahir Al-Baghdadi ( 980 - 1037 ), René Descartes ( 1596 - 1650 ), to whom the formula of Thabit is sometimes ascribed, C. Rudolphus and others. Thabit's formula was generalized by Euler . If a number equals the sum of ''its own'' proper divisors, it is called a Perfect Number . SEE ALSO EXTERNAL LINKS
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