Editing Amicable numbers (section)

== History ==
{{unsolved|mathematics|Are there infinitely many amicable numbers?}}
Amicable numbers were known to the [[Pythagoreanism|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 people|Iraqi]] mathematician [[Thābit ibn Qurra]] (826–901). Other [[Arab]] mathematicians who studied amicable numbers are [[Maslamah Ibn Ahmad al-Majriti|al-Majriti]] (died 1007), [[Ibn Tahir al-Baghdadi|al-Baghdadi]] (980–1037), and [[Kamāl al-Dīn al-Fārisī|al-Fārisī]] (1260–1320). The [[Iran]]ian mathematician [[Muhammad Baqir Yazdi]] (16th century) discovered the pair (9363584, 9437056), though this has often been attributed to [[René Descartes|Descartes]].<ref>{{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 }}</ref> Much of the work of [[Mathematics in medieval Islam|Eastern mathematicians]] in this area has been forgotten.

Thābit ibn Qurra's formula was rediscovered by [[Pierre de Fermat|Fermat]] (1601–1665) and [[René Descartes|Descartes]] (1596–1650), to whom it is sometimes ascribed, and extended by [[Leonhard Euler|Euler]] (1707–1783). It was extended further by [[Walter Borho|Borho]] in 1972. Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs.<ref name=Sandifer/> 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.<ref>{{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}}</ref><Ref>
{{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}}
</ref>

{| class="wikitable sortable"
|+ 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
|}

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