Amicable numbers

{{unsolved|mathematics|Are there infinitely many amicable numbers?}}

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.{{cite journal | last = Costello | first = Patrick | title = New Amicable Pairs Of Type (2; 2) And Type (3; 2) | journal = Mathematics of Computation | volume = 72 | pages = 489–497 | date = 1 May 2002 | url = https://www.ams.org/mcom/2003-72-241/S0025-5718-02-01414-X/S0025-5718-02-01414-X.pdf | access-date = 19 April 2007 | issue = 241 | doi = 10.1090/S0025-5718-02-01414-X | archive-date = 2008-02-29 | archive-url = https://web.archive.org/web/20080229172358/http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01414-X/S0025-5718-02-01414-X.pdf | url-status = live }} 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 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.{{cite web|last=Sprugnoli|first=Renzo|title=Introduzione alla matematica: La matematica della scuola media|url=http://www.dsi.unifi.it/~resp/media.pdf|publisher=Universita degli Studi di Firenze: Dipartimento di Sistemi e Informatica|access-date=21 August 2012|page=59|language=it|date=27 September 2005|url-status=dead|archive-url=https://web.archive.org/web/20120913033238/http://www.dsi.unifi.it/~resp/media.pdf|archive-date=13 September 2012}}

{{cite book|url=https://books.google.com/books?id=kE0FEAAAQBAJ&dq=Nicol%C3%B2+I.+Paganini+mathematician&pg=PA168|page=168|author=Martin Gardner|title=Mathematical Magic Show|orig-date=Originally published in 1977|date=2020|publisher=American Mathematical Society|isbn=9781470463588|access-date=2023-03-18|archive-date=2023-09-12|archive-url=https://web.archive.org/web/20230912194538/https://books.google.com/books?id=kE0FEAAAQBAJ&dq=Nicol%C3%B2+I.+Paganini+mathematician&pg=PA168|url-status=live}}

There are over 1 billion known amicable pairs.{{cite web|first=Sergei|last=Chernykh|url=http://sech.me/ap/|title=Amicable pairs list|access-date=2024-05-28}}

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)].

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

It states that if

$$\backslash begin\{align\}$$

p &= 3 \times 2^{n-1} - 1, \\

q &= 3 \times 2^{n} - 1, \\

r &= 9 \times 2^{2n - 1} - 1,

\end{align}

where {{math|n > 1}} is an integer and {{mvar|p, q, r}} are prime numbers, then {{math|2ⁿ × p × q}} and {{math|2ⁿ × r}} are a pair of amicable numbers. This formula gives the pairs {{math|(220, 284)}} for {{math|n {{=}} 2}}, {{math|(17296, 18416)}} for {{math|n {{=}} 4}}, and {{math|(9363584, 9437056)}} for {{math|n {{=}} 7}}, but no other such pairs are known. Numbers of the form {{math|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 {{mvar|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.{{cite book|last=Rashed|first=Roshdi|title=The development of Arabic mathematics: between arithmetic and algebra.|publisher=Kluwer Academic Publishers|location=Dordrecht, Boston, London|year=1994|volume=156|isbn=978-0-7923-2565-9|page=278,279}}

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

$$\backslash begin\{align\}$$

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{align}

where {{math|n > m > 0}} are integers and {{mvar|p, q, r}} are prime numbers, then {{math|2ⁿ × p × q}} and {{math|2ⁿ × r}} are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case {{math|m {{=}} n − 1}}. Euler's rule creates additional amicable pairs for {{math|(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.{{cite book | title=How Euler Did It | last=Sandifer | first=C. Edward | isbn=978-0-88385-563-8 | pages=49–55 | year=2007 | publisher=Mathematical Association of America }}See William Dunham in a video: [https://www.youtube.com/watch?v=h-DV26x6n_Q&t=37m An Evening with Leonhard Euler – YouTube] {{Webarchive|url=https://web.archive.org/web/20160516062654/https://www.youtube.com/watch?v=h-DV26x6n_Q&t=37m |date=2016-05-16 }}

Let ({{mvar|m}}, {{mvar|n}}) be a pair of amicable numbers with {{math|m < n}}, and write {{math|m {{=}} gM}} and {{math|n {{=}} gN}} where {{mvar|g}} is the greatest common divisor of {{mvar|m}} and {{mvar|n}}. If {{mvar|M}} and {{mvar|N}} are both coprime to {{mvar|g}} and square free then the pair ({{mvar|m}}, {{mvar|n}}) is said to be regular {{OEIS|A215491}}; otherwise, it is called irregular or exotic. If ({{mvar|m}}, {{mvar|n}}) is regular and {{mvar|M}} and {{mvar|N}} have {{mvar|i}} and {{mvar|j}} prime factors respectively, then {{math|(m, n)}} is said to be of type {{math|(i, j)}}.

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

An amicable pair {{math|(m, n)}} is twin if there are no integers between {{mvar|m}} and {{mvar|n}} belonging to any other amicable pair {{OEIS|A273259}}.

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.{{Cite web|url=http://sech.me/ap/news.html#20160130|title=Amicable pairs news|access-date=2016-01-31|archive-date=2021-07-18|archive-url=https://web.archive.org/web/20210718213137/https://sech.me/ap/news.html#20160130|url-status=live}} 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.{{cite journal

| last = Hagis | first = Peter, Jr.

| doi = 10.2307/2004381

| journal = Mathematics of Computation

| mr = 246816

| pages = 539–543

| title = On relatively prime odd amicable numbers

| volume = 23

| year = 1969| issue = 107

| jstor = 2004381

}}{{cite journal

| last = Hagis | first = Peter, Jr.

| doi = 10.2307/2004629

| journal = Mathematics of Computation

| mr = 276167

| pages = 963–968

| title = Lower bounds for relatively prime amicable numbers of opposite parity

| volume = 24

| year = 1970| issue = 112

| jstor = 2004629

}} Also, a pair of co-prime 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.{{cite journal|last1=Erdős|first1=Paul|title=On amicable numbers|journal=Publicationes Mathematicae Debrecen|year=2022 |volume=4|issue=1–2 |pages=108–111|doi=10.5486/PMD.1955.4.1-2.16 |s2cid=253787916 |url=https://www.renyi.hu/~p_erdos/1955-03.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.renyi.hu/~p_erdos/1955-03.pdf |archive-date=2022-10-09 |url-status=live}}

In 1968 Martin Gardner noted that most even amicable pairs have sums divisible by 9,{{Cite journal|last=Gardner|first=Martin|title=Mathematical Games|date=1968|url=https://www.jstor.org/stable/24926005|journal=Scientific American|volume=218|issue=3|pages=121–127|doi=10.1038/scientificamerican0368-121|jstor=24926005|bibcode=1968SciAm.218c.121G|issn=0036-8733|access-date=2020-09-07|archive-date=2022-09-25|archive-url=https://web.archive.org/web/20220925113302/https://www.jstor.org/stable/24926005|url-status=live}} and that a rule for characterizing the exceptions {{OEIS|A291550}} was obtained.{{Cite journal|last=Lee|first=Elvin|date=1969|title=On Divisibility by Nine of the Sums of Even Amicable Pairs|journal=Mathematics of Computation|volume=23|issue=107|pages=545–548|doi=10.2307/2004382|jstor=2004382|issn=0025-5718|doi-access=free}}

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% {{OEIS|A291422}}. 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 the even amicable pairs (A360054 in OEIS).

Gaussian integer amicable pairs exist,Patrick Costello, Ranthony A. C. Edmonds. "Gaussian Amicable Pairs." Missouri Journal of Mathematical Sciences, 30(2) 107-116 November 2018.{{Cite journal|url=https://encompass.eku.edu/etd/158/|title=Gaussian Amicable Pairs|first=Ranthony|last=Clark|date=January 1, 2013|journal=Online Theses and Dissertations}} e.g.

s(8008+3960i) = 4232-8280i and s(4232-8280i) = 8008+3960i.{{Cite web|url=https://mathworld.wolfram.com/AmicablePair.html|title=Amicable Pair|first=Eric W.|last=Weisstein|website=mathworld.wolfram.com}}

Amicable numbers $(m,\; n)$ satisfy $\backslash sigma(m)-m=n$ and $\backslash sigma(n)-n=m$ which can be written together as $\backslash sigma(m)=\backslash sigma(n)=m+n$. This can be generalized to larger tuples, say $(n\_1,n\_2,\backslash ldots,n\_k)$, where we require

:$\backslash sigma(n\_1)=\backslash sigma(n\_2)=\; \backslash dots\; =\backslash sigma(n\_k)\; =\; n\_1+n\_2+\; \backslash dots\; +n\_k$

For example, (1980, 2016, 2556) is an amicable triple {{OEIS|id=A125490}}, and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple {{OEIS|id=A036471}}.

Amicable multisets are defined analogously and generalizes this a bit further {{OEIS|id=A259307}}.

{{main article|Sociable number}}

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, $1264460\; \backslash mapsto\; 1547860\; \backslash mapsto\; 1727636\; \backslash mapsto\; 1305184\; \backslash mapsto\; 1264460\; \backslash mapsto\backslash dots$ are sociable numbers of order 4.

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 $k$.{{citation|title=Distributed cycle detection in large-scale sparse graphs|first1=Rodrigo Caetano|last1=Rocha|first2=Bhalchandra|last2=Thatte|year=2015|publisher=Simpósio Brasileiro de Pesquisa Operacional (SBPO)|doi=10.13140/RG.2.1.1233.8640}}

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.

• 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.

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

{{reflist}}

{{Wikisource1911Enc|Amicable Numbers}}

• {{EB1911|wstitle=Amicable Numbers}}
• {{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=978-1-4020-2546-4 | pages=32–36 | zbl=1079.11001 }}
• {{cite book | last=Wells | first=D. | year=1987 | title=The Penguin Dictionary of Curious and Interesting Numbers | pages=145–147 | location=London | publisher=Penguin Group | title-link=The Penguin Dictionary of Curious and Interesting Numbers }}
• {{MathWorld |title=Amicable Pair|urlname=AmicablePair}}
• {{MathWorld |title=Thâbit ibn Kurrah Rule|urlname=ThabitibnKurrahRule|author =Weisstein, Eric W.}}
• {{MathWorld |title=Euler's Rule|urlname=EulersRule|author =Weisstein, Eric W.}}

• {{cite journal |author1=M. García |author2=J.M. Pedersen |author3=H.J.J. te Riele |title=Amicable pairs, a survey |journal=Report MAS-R0307 |date=2003-07-31 |url=http://oai.cwi.nl/oai/asset/4143/04143D.pdf}}
• {{cite web|last=Grime|first=James|title=220 and 284 (Amicable Numbers)|url=http://www.numberphile.com/videos/220_284.html|work=Numberphile|publisher=Brady Haran|access-date=2013-04-02|archive-url=https://web.archive.org/web/20170716184637/http://www.numberphile.com/videos/220_284.html|archive-date=2017-07-16|url-status=dead}}
• {{cite web|last=Grime|first=James|title=MegaFavNumbers - The Even Amicable Numbers Conjecture|url=https://www.youtube.com/watch?v=R2eQVqdUQLI| archive-url=https://ghostarchive.org/varchive/youtube/20211123/R2eQVqdUQLI| archive-date=2021-11-23 | url-status=live|work=YouTube|access-date=2020-06-09}}{{cbignore}}
• {{cite web|last=Koutsoukou-Argyraki|first=Angeliki|title=Amicable Numbers (Formal proof development in Isabelle/HOL, Archive of Formal Proofs) |date=4 August 2020 |url=https://www.isa-afp.org/entries/Amicable_Numbers.html}}
• {{cite web|first=Sergei|last=Chernykh|url=https://sech.me/ap/|title=Amicable pairs list|access-date=2023-09-10}} (database of all known amicable numbers)

{{Divisor classes}}

{{Classes of natural numbers}}

{{DEFAULTSORT:Amicable Number}}

Category:Arithmetic dynamics

Category:Divisor function

Category:Integer sequences