Amicable numbers

"Amicable" redirects here. For other uses, see wikt:amicable.

Not to be confused with friendly numbers.

Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence of period 2. 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.

For example, the smallest pair of amicable numbers 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.

The first 20 amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), (66928, 66992), (67095, 71145), (69615, 87633), (79750, 88730), (100485, 124155), (122265, 139815), (122368, 123152), (141664, 153176), (142310, 168730), ... (sequence A259180 in OEIS). (Also see  A002025 and  A002046)

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 (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. 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). 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, (1184, 1210), was discovered in 1866 by a then teenage B. Nicolò I. Paganini (no to be confused with the composer and violinist), 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, and to 10¹⁷ in 2015.

As of December 2015, there are 40,871,144 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.

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

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 2ⁿ×p×q and 2ⁿ×r are a pair of amicable numbers. This formula gives the pairs (220, 284) for n=2, (17296, 18416) for n=4, and (9363584, 9437056) for n=7, but no other such pairs are known. Numbers of the form 3 × 2ⁿ − 1 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 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.

Euler's rule

Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if

p = (2^(n - m)+1) × 2^m − 1,

q = (2^(n - m)+1) × 2ⁿ − 1,

r = (2^(n - m)+1)² × 2^m + n − 1,

where n > m > 0 are integers and p, q, and r are prime numbers, then 2ⁿ×p×q and 2ⁿ×r are a pair of amicable numbers. 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. William Dunham in a video claims that Euler (1750) found 58 such pairs to make all the by then existing pairs 61.

Regular pairs

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, otherwise it is called irregular or exotic. If (m, n) is regular and M and N have i and j prime factors respectively, then (m, n) is said to be of type (i, j).

For example, with (m, n) = (220, 284), the greatest common divisor is 4 and so M = 55 and N = 71. Therefore (220, 284) is regular of type (2, 1).

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. Also, every known pair shares at least one common factor, higher than 1. 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 (above), nor by any similar formula.

In 1955, Paul Erdős showed that the density of amicable numbers, relative to the positive integers, was 0.

References in popular culture

• Amicable numbers are featured in the novel The Professor's Beloved Equation by Yoko Ogawa, and in the Japanese film based on it.
• Paul Auster's collection of short stories entitled True Tales of American Life contains a story ('Mathematical Aphrodisiac' by Alex Galt) in which amicable numbers play an important role.
• Amicable numbers are featured briefly in the novel The Stranger House by Reginald Hill.
• Amicable numbers are mentioned in the French novel The Parrot's Theorem by Denis Guedj.
• Amicable numbers are featured in the visual novel Rewrite (visual novel).

Generalizations

Amicable tuples

Amicable numbers $(m, n)$ satisfy $\sigma(m)-m=n$ and $\sigma(n)-n=m$ which can be written together as $\sigma(m)=\sigma(n)=m+n$. This can be generalized to larger tuples, say $(n_1,n_2,\ldots,n_k)$, where we require

$\sigma(n_1)=\sigma(n_2)= \dots =\sigma(n_k) = n_1+n_2+ \dots +n_k$

For example (1980, 2016, 2556) is an amicable triple (sequence A125490 in OEIS), and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple (sequence A036471 in OEIS).

Amicable multisets are defined analogously and generalizes this a bit further (sequence A259307 in OEIS).

Sociable numbers

Main article: sociable number

Sociable numbers are a cyclic lists of numbers such that each number is the sum of the proper divisors of the preceding number. For example $1264460 \mapsto 1547860 \mapsto 1727636 \mapsto 1305184 \mapsto 1264460 \mapsto\dots$ are sociable numbers of order 4.

Searching for sociable numbers

The aliquot sequence can be represented as a directed graph, $G_{n,s}$, for a given integer $n$, where $s(k)$ denotes the sum of the proper divisors of $k$. Cycles in $G_{n,s}$ represent sociable numbers within the interval $[1,n]$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

See also

• Betrothed numbers (quasi-amicable numbers)

References

•  This article incorporates text from a publication now in the public domain: 
• Weisstein, Eric W., "Amicable Pair", MathWorld.
• Weisstein, Eric W., "Thâbit ibn Kurrah Rule", MathWorld.
• Weisstein, Eric W., "Euler's Rule", MathWorld.