Editing Amicable numbers (section)

== 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 [[Arab]] [[mathematician]] [[Thābit ibn Qurrah]].<ref name="Rashed"/>

It states that if
<math display=block>\begin{align}
  p &= 3 \times 2^{n-1} - 1, \\
  q &= 3 \times 2^{n} - 1, \\
  r &= 9 \times 2^{2n - 1} - 1,
\end{align}</math>

where {{math|''n'' > 1}} is an [[integer]] and {{mvar|p, q, r}} are [[prime number]]s, then {{math|2<sup>''n''</sup> × ''p'' × ''q''}} and {{math|2<sup>''n''</sup> × ''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 × 2<sup>''n''</sup> − 1}} are known as [[Thabit number]]s. 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 [[Lemma (mathematics)|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.<ref name="Rashed">{{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}}</ref>

===Euler's rule===
''Euler's rule'' is a generalization of the Thâbit ibn Qurra theorem. It states that if
<math display=block>\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}</math>
where {{math|''n'' > ''m'' > 0}} are [[integer]]s and {{mvar|p, q, r}} are [[prime number]]s, then {{math|2<sup>''n''</sup> × ''p'' × ''q''}} and {{math|2<sup>''n''</sup> × ''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.<ref name=Sandifer>{{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]] }}</ref><ref>See [[William Dunham (mathematician)|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 }}</ref>