Editing Amicable numbers (section)

===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>