Amicable numbers

Amicable numbers are two different

).

The smallest pair of amicable numbers is (220, 284). 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 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.)

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). (Also see OEIS: A002025 and OEIS: A002046) It is unknown if there are infinitely many pairs of amicable numbers.

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.

History

Amicable numbers were known to the

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.^[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.^[3][4]

The First Ten Amicable Pairs

#	m	n
1	220	284
2	1,184	1,210
3	2,620	2,924
4	5,020	5,564
5	6,232	6,368
6	10,744	10,856
7	12,285	14,595
8	17,296	18,416
9	63,020	76,084
10	66,928	66,992

As of 15 April 2024^[update], there are over 1,228,940,050 known amicable pairs.^[5]

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 [García, Pedersen & te Riele (2003), Sándor & Crstici (2004)].

Thābit ibn Qurrah theorem

The Thābit ibn Qurrah theorem is a method for discovering amicable numbers invented in the 9th century by the

It states that if

$${\displaystyle {\begin{aligned}p&=3\times 2^{n-1}-1,\\q&=3\times 2^{n}-1,\\r&=9\times 2^{2n-1}-1,\end{aligned}}}$$

where n > 1 is an integer and p, q, 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 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

Euler's rule

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

$${\displaystyle {\begin{aligned}p&=(2^{n-m}+1)\times 2^{m}-1,\\q&=(2^{n-m}+1)\times 2^{n}-1,\\r&=(2^{n-m}+1)^{2}\times 2^{m+n}-1,\end{aligned}}}$$

where n > m > 0 are integers and p, q, 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. Euler (1747 & 1750) overall found 58 new pairs increasing the number of pairs that were then known to 61.^[2][7]

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

); 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).

Twin amicable pairs

An amicable pair (m, n) is twin if there are no integers between m and n belonging to any other amicable pair (sequence A273259 in the OEIS).

Other results

In every known case, the numbers of a pair are either both

] 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.^[9]

In 1968, Martin Gardner noted that most even amicable pairs known at his time have sums divisible by 9,^[10] and a rule for characterizing the exceptions (sequence A291550 in the OEIS) was obtained.^[11]

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). Although all amicable pairs up to 10,000 are even pairs, the proportion of odd amicable pairs increases steadily towards higher numbers, and presumably there are more of them than of even amicable pairs (A360054 in OEIS).

Gaussian amicable pairs exist.^[12]

Generalizations

Amicable tuples

Amicable numbers ${\displaystyle (m,n)}$ satisfy ${\displaystyle \sigma (m)-m=n}$ and ${\displaystyle \sigma (n)-n=m}$ which can be written together as ${\displaystyle \sigma (m)=\sigma (n)=m+n}$. This can be generalized to larger tuples, say ${\displaystyle (n_{1},n_{2},\ldots ,n_{k})}$, where we require

${\displaystyle \sigma (n_{1})=\sigma (n_{2})=\dots =\sigma (n_{k})=n_{1}+n_{2}+\dots +n_{k}}$

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

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

Sociable numbers

Sociable numbers are the numbers in cyclic lists of numbers (with a length greater than 2) where each number is the sum of the proper divisors of the preceding number. For example, ${\displaystyle 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto \dots }$ are sociable numbers of order 4.

Searching for sociable numbers

The aliquot sequence can be represented as a directed graph, ${\displaystyle G_{n,s}}$, for a given integer ${\displaystyle n}$, where ${\displaystyle s(k)}$ denotes the sum of the proper divisors of ${\displaystyle k}$.^[13] Cycles in ${\displaystyle G_{n,s}}$ represent

within the interval ${\displaystyle [1,n]}$. Two special cases are loops that represent .

References in popular culture

• Amicable numbers are featured in the novel The Housekeeper and the Professor by Yōko 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 mentioned in the JRPG
  Persona 4 Golden
  .
• Amicable numbers are featured in the visual novel
  Rewrite
  .
• Amicable numbers (220, 284) are referenced in episode 13 of the 2017 Korean drama Andante.
• Amicable numbers are featured in the Greek movie The Other Me (2016 film).
• Amicable numbers are discussed in the book Are Numbers Real? by Brian Clegg.
• Amicable numbers are mentioned in the 2020 novel Apeirogon by Colum McCann.

See also

• Betrothed numbers (quasi-amicable numbers)
• Amicable triple - Three-number variation of Amicable numbers.

Notes

References

Wikisource has the text of the 1911 Encyclopædia Britannica article "Amicable Numbers".

•  This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Amicable Numbers". Encyclopædia Britannica (11th ed.). Cambridge University Press.
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36.
  Zbl 1079.11001
  .
• Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Group. pp. 145–147.
• Weisstein, Eric W. "Amicable Pair". MathWorld.
• Weisstein, Eric W. "Thâbit ibn Kurrah Rule". MathWorld.
• Weisstein, Eric W. "Euler's Rule". MathWorld.

External links

• M. García; J.M. Pedersen; H.J.J. te Riele (2003-07-31). "Amicable pairs, a survey" (PDF). Report MAS-R0307.
• Grime, James. "220 and 284 (Amicable Numbers)". Numberphile. Brady Haran. Archived from the original on 2017-07-16. Retrieved 2013-04-02.
• Grime, James. "MegaFavNumbers - The Even Amicable Numbers Conjecture". YouTube. Archived from the original on 2021-11-23. Retrieved 2020-06-09.
• Koutsoukou-Argyraki, Angeliki (4 August 2020). "Amicable Numbers (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)".
• Chernykh, Sergei. "Amicable pairs list". Retrieved 2023-09-10. (database of all known amicable numbers)