Amicable numbers


Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number.
The smallest pair of amicable numbers is. They are amicable because 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.
A pair of amicable numbers constitutes an aliquot sequence of period 2. It is unknown if there are infinitely many pairs of amicable numbers.
A related concept is that of a perfect number, which is a number that equals the sum of its own proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers.
The first ten amicable pairs are:,,,,,,,,, and..

History

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. Other Arab mathematicians who studied amicable numbers are al-Majriti, al-Baghdadi, and al-Fārisī. The Iranian mathematician Muhammad Baqir Yazdi discovered the pair, though this has often been attributed to Descartes. Much of the work of Eastern mathematicians in this area has been forgotten.
Thābit ibn Qurra's formula was rediscovered by Fermat and Descartes, to whom it is sometimes ascribed, and extended by Euler. It was extended further by Borho in 1972. Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs. The second smallest pair,, was discovered in 1866 by a then teenage B. Nicolò I. Paganini, having been overlooked by earlier mathematicians.
By 1946 there were 390 known pairs, but the advent of computers has allowed the discovery of many thousands since then. Exhaustive searches have been carried out to find all pairs less than a given bound, this bound being extended from 108 in 1970, to 1010 in 1986, 1011 in 1993, 1017 in 2015, and to 1018 in 2016.
, there are over 1,225,063,681 known amicable pairs.

Rules for generation

While these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive.
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 .

Thābit ibn Qurra theorem

The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the ninth century by the Arab mathematician Thābit ibn Qurra.
It states that if
where is an integer and,, and are prime numbers, then and are a pair of amicable numbers. This formula gives the pairs for, for, and for, but no other such pairs are known. Numbers of the form are known as Thabit numbers. In order for Ibn Qurra's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of.
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.

Euler's rule

Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if
where are integers and,, and are prime numbers, then and are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case. Euler's rule creates additional amicable pairs for with no others being known. Euler overall found 58 new pairs to make all the by then existing pairs into 61.

Regular pairs

Let be a pair of amicable numbers with, and write and where is the greatest common divisor of and. If and are both coprime to and square free then the pair is said to be regular, otherwise it is called irregular or exotic. If is regular and and have and prime factors respectively, then is said to be of type.
For example, with, the greatest common divisor is and so and. Therefore, is regular of type.

Twin amicable pairs

An amicable pair is twin if there are no integers between and belonging to any other amicable pair.

Other results

In every known case, the numbers of a pair are either both even or both odd. It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known. 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 1067. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula, nor by any similar formula.
In 1955, Paul Erdős showed that the density of amicable numbers, relative to the positive integers, was 0.
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%.

Generalizations

Amicable tuples

Amicable numbers satisfy and which can be written together as. This can be generalized to larger tuples, say, where we require
For example, is an amicable triple, and is an amicable quadruple.
Amicable multisets are defined analogously and generalizes this a bit further.

Sociable numbers

Sociable numbers are the numbers in cyclic lists of numbers where each number is the sum of the proper divisors of the preceding number. For example, are sociable numbers of order 4.

Searching for sociable numbers

The aliquot sequence can be represented as a directed graph,, for a given integer, where denotes the
sum of the proper divisors of.
Cycles in represent sociable numbers within the interval. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.