Amicable numbers
The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992) (sequence A259180 in the OEIS).
It is unknown if there are infinitely many pairs of amicable numbers.
A pair of amicable numbers constitutes an aliquot sequence of period 2.
Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties.
A general formula by which some of these numbers could be derived was invented circa 850 by the Iraqi mathematician Thābit ibn Qurra (826–901).
Other Arab mathematicians who studied amicable numbers are al-Majriti (died 1007), al-Baghdadi (980–1037), and al-Fārisī (1260–1320).
The Iranian mathematician Muhammad Baqir Yazdi (16th century) discovered the pair (9363584, 9437056), though this has often been attributed to Descartes.
[1] Much of the work of Eastern mathematicians in this area has been forgotten.
Thābit ibn Qurra's formula was rediscovered by Fermat (1601–1665) and Descartes (1596–1650), to whom it is sometimes ascribed, and extended by Euler (1707–1783).
Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians.
[2] The second smallest pair, (1184, 1210), was discovered in 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.
In particular, the two rules below produce only even amicable pairs, so they are of no interest for the open problem of finding amicable pairs coprime to 210 = 2·3·5·7, while over 1000 pairs coprime to 30 = 2·3·5 are known [García, Pedersen & te Riele (2003), Sándor & Crstici (2004)].
The Thābit ibn Qurrah theorem is a method for discovering amicable numbers invented in the 9th century by the Arab mathematician Thābit ibn Qurrah.
In order for Ibn Qurrah's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of n. To establish the theorem, Thâbit ibn Qurra proved nine lemmas divided into two groups.
The first three lemmas deal with the determination of the aliquot parts of a natural integer.
The second group of lemmas deals more specifically with the formation of perfect, abundant and deficient numbers.
[6] Euler's rule is a generalization of the Thâbit ibn Qurra theorem.
Thābit ibn Qurra's theorem corresponds to the case m = n − 1.
Euler's rule creates additional amicable pairs for (m,n) = (1,8), (29,40) with no others being known.
[2][7] Let (m, n) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair (m, n) is said to be regular (sequence A215491 in the OEIS); otherwise, it is called irregular or exotic.
However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known.
[8] Also, every known pair shares at least one common prime factor.
It is not known whether a pair of coprime amicable numbers exists, though if any does, the product of the two must be greater than 1065.
In 1955 Paul Erdős showed that the density of amicable numbers, relative to the positive integers, was 0.
[11] In 1968 Martin Gardner noted that most even amicable pairs sumsdivisible by 9,[12] and that a rule for characterizing the exceptions (sequence A291550 in the OEIS) was obtained.
[13] According to the sum of amicable pairs conjecture, as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100% (sequence A291422 in the OEIS).
Gaussian integer amicable pairs exist,[14][15] e.g. s(8008+3960i) = 4232-8280i and s(4232-8280i) = 8008+3960i.
Amicable multisets are defined analogously and generalizes this a bit further (sequence A259307 in the OEIS).
The aliquot sequence can be represented as a directed graph,
