History In about 300 BC Euclid showed that if 2p − 1 is prime then 2p−1(2p − 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 100. In modern language, Nicomachus states without proof that perfect number is of the form where is prime.In 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. He seems to be unaware that 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,Commentary on the Gospel of John 28.1.1–4, with further references in the Sources Chrétiennes edition: vol. 385, 58–61. 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). St Augustine defines perfect numbers in 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 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.Roshdi Rashed, The Development of Arabic Mathematics: Between Arithmetic and Algebra (Dordrecht: Kluwer Academic Publishers, 1994), pp. 328–329. The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.Bayerische Staatsbibliothek, Clm 14908. See 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.
hiStory in aBout 300 bc EucliD showed that IF 2p − 1 iS Prime TheN 2P−1(2p − 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 100. In moderN lanGUage, NicoMaChus STAtes without pRoOF thaT perfECt NuMber is Of tHE foRm where IS priMe.in 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. hE sEEMs to bE unaware tHaT 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,coMmENTarY oN the gospel Of jOhn 28.1.1–4, WiTh FUrthEr ReferenceS in thE SOUrCes cHrÉtIEnnEs eDitIoN: VoL. 385, 58–61. 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). st augusTine defIneS PeRFecT NumbErs in 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 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.ROsHdi rAshed, THe dEveLOpmeNt of arABiC mathEmatiCs: betwEEn arIthmetIc And alGEbrA (dordReCht: kluwer Academic publiSheRs, 1994), pp. 328–329. tHe fIrST KNOWn european mention oF the fIFth perFecT nUmBeR is a mANUSCriPT wrItten betwEeN 1456 and 1461 by an uNkNOwn mAthEmatICiAn.baYeRiSche staaTsBIblioTHEk, clm 14908. see 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.