Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. Such a pair is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors 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 number is an abstract entity used originally to describe quantity. ... Addition (or summation) is one of the basic operations of arithmetic. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Look up one on Wiktionary, the free dictionary. ... 220 (two hundred [and] twenty) is the natural number following 219 and preceding 221. ... Two hundred eighty-four (284, CCLXXXIV) is the natural number following 283 and preceding 285. ... The Pythagoreans were a Hellenic organization of astronomers, musicians, mathematicians, and philosophers who believed that all things are, essentially, numeric. ...

A general formula by which these numbers could be derived was invented circa 850 by Thabit ibn Qurra (826-901): if Events April 20 - Guntherus becomes Bishop of Cologne. ... Abul Hasan Thabit ibn Qurra ibn Marwan al-Sabi al-Harrani, (826 – February 18, 901) was an Arab astronomer and mathematician. ... Events The Danish king accepts Christianity. ... Events Mesoamerican ballgame court dedicated at Uxmal Kingdom of Taebong established in Korean peninsula Fuzhou city was expanded with construction of a new city wall (Luo City). Births Deaths February 18 - Thabit ibn Qurra, Arab astronomer and mathematician Categories: 901 ...

p = 3 × 2^n-1 - 1,

q = 3 × 2ⁿ - 1,

r = 9 × 2^2n-1 - 1,

where n > 1 is an integer and p, q, and r are prime numbers, then 2ⁿpq and 2ⁿr 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. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

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. In mathematics, the term even is used in several senses: an integer is called even if it is divisible by two; see even and odd numbers a function f defined on the real numbers is called even if f(x) = f(−x) for all x; see even and odd functions... Odd has several meanings. ... A factor can be: a person acting as a mercantile agent, or, in Scotland, a Factor is a person who manages property on behalf of the owner; in mathematics, a multiplicative factor is a synonym for coefficient a number that is a divisor of another number; thus, factorization is to... Coprime - Wikipedia /**/ @import /skins-1. ...

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 formula was generalized by Euler. Events Aethelred buys two years of peace with the Danes for 36,000 pounds of silver. ... Events Births Emperor Ichijo of Japan Humbert I of Savoy Avicenna Godiva, Countess of Mercia Deaths Categories: 980 ... // Events Construction of the church of Saint Sophia Cathedral is started in Kyiv. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Events February 5 - 26 catholics crucified in Nagasaki, Japan. ... // Events June 23 - Claimant King Charles II of England, Scotland and Ireland arrives in Scotland, the only of the three Kingdoms that has accepted him as ruler. ... Leonhard Euler by Emanuel Handmann Leonhard Euler [oilÉ™r] (April 15, 1707–September 18, 1783) was a Swiss mathematician and physicist. ...

If a number equals the sum of its own proper divisors, it is called a perfect number. In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself. ...

See also

• Sociable numbers

Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ...

External links

• All known amicable numbers