Editing Perfect number (section)

== History ==
In about 300&nbsp;BC Euclid showed that if 2<sup>''p''</sup>&nbsp;−&nbsp;1 is prime then 2<sup>''p''−1</sup>(2<sup>''p''</sup>&nbsp;−&nbsp;1) is perfect.
The first four perfect numbers were the only ones known to early [[Greek mathematics]], and the mathematician [[Nicomachus]] noted 8128 as early as around AD&nbsp;100.<ref name="Dickinson LE (1919)">{{cite book|last=Dickson|first=L. E. | author-link = L. E. Dickson|title=History of the Theory of Numbers, Vol. I|page=4|year=1919|publisher=Carnegie Institution of Washington|location=Washington|url=https://archive.org/stream/historyoftheoryo01dick#page/4/}}</ref> In modern language, Nicomachus states without proof that {{em|every}} perfect number is of the form <math>2^{n-1}(2^n-1)</math> where <math>2^n-1</math> is prime.<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Perfect_numbers.html|title=Perfect numbers|website=www-groups.dcs.st-and.ac.uk|access-date=9 May 2018}}</ref><ref>In ''[https://archive.org/download/NicomachusIntroToArithmetic/nicomachus_introduction_arithmetic.pdf Introduction to Arithmetic]'', Chapter 16, he says of perfect numbers, "There is a method of producing them, neat and unfailing, which neither passes by any of the perfect numbers nor fails to differentiate any of those that are not such, which is carried out in the following way." He then goes on to explain a procedure which is equivalent to finding a [[triangular number]] based on a Mersenne prime.</ref> He seems to be unaware that {{mvar|n}} itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.) [[Philo of Alexandria]] in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by [[Origen]],<ref>Commentary on the Gospel of John 28.1.1–4, with further references in the [[Sources Chrétiennes]] edition: vol. 385, 58–61.</ref> and by [[Didymus the Blind]],  who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19).<ref>{{cite conference|url=http://torreys.org/sblpapers2015/S22-05_philonic_arithmological_exegesis.pdf |first=Justin M.|last=Rogers|title=The Reception of Philonic Arithmological Exegesis in Didymus the Blind's ''Commentary on Genesis''|work=Society of Biblical Literature National Meeting, Atlanta, Georgia|year=2015}}</ref> [[Augustine of Hippo]] defines perfect numbers in ''[[The City of God]]'' (Book XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician [[Ibn Fallus|Ismail ibn Fallūs]] (1194–1252) mentioned the next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) and listed a few more which are now known to be incorrect.<ref>Roshdi Rashed, ''The Development of Arabic Mathematics: Between Arithmetic and Algebra'' (Dordrecht: Kluwer Academic Publishers, 1994), pp.&nbsp;328–329.</ref> The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.<ref>[[Bayerische Staatsbibliothek]], Clm 14908. See {{cite book|author=David Eugene Smith|author-link=David Eugene Smith|title=History of Mathematics: Volume II|year=1925|publisher=Dover|location=New York|isbn=0-486-20430-8|pages=21|url=https://archive.org/stream/historyofmathema031897mbp#page/n35/mode/2up}}</ref> In 1588, the Italian mathematician [[Pietro Cataldi]] identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.<ref>{{cite book|last=Dickson|first=L. E. | author-link = L. E. Dickson|title=History of the Theory of Numbers, Vol. I|year=1919|publisher=Carnegie Institution of Washington|location=Washington|page=10|url=https://archive.org/stream/historyoftheoryo01dick#page/10/}}</ref><ref name="Pickover C (2001)">{{cite book|last=Pickover|first=C|title=Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning|year=2001|publisher=Oxford University Press|location=Oxford|isbn=0-19-515799-0|pages=360|url=https://books.google.com/books?id=52N0JJBspM0C&pg=PA360}}</ref><ref name="Peterson I (2002)">{{cite book|last=Peterson|first=I|title=Mathematical Treks: From Surreal Numbers to Magic Circles|year=2002|publisher=Mathematical Association of America|location=Washington|isbn=88-8358-537-2|pages=132|url=https://books.google.com/books?id=4gWSAraVhtAC&pg=PA132}}</ref>