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Простые числа Мерсенна. Совершенные числа

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I?inoua *enea Ia?naiia, niaa?oaiiua /enea.

N?aaee i?inouo /enae iniaoth ?ieue ea?atho i?inoua /enea Ia?naiia –
/enea aeaea 1)I? = 2? -1 , aaea ? – i?inoia /enei. Iie iacuaathony
i?inouie /eneaie Ia?naiia ii eiaie o?aioeocneiai iiiaoa Ia?aia Ia?naiia
(1588-1648), iaeiiai ec iniiaaoaeae Ia?eaeneie Aeaaeaiee iaoe, ae?oaa
Aeaea?oa e Oa?ia. Oae eae I2=3, I3=7, I5=31, I7=127, oi yoi – i?inoua
/enea Ia?naiia. Iaeiaei, /enei 2)I11=2047=23 . 89 i?inoui ia
yaeyaony. Aei 1750 aiaea auei iaeaeaii anaai 8 i?inouo /enae Ia?naiia:
I2, I3, I5, I7, I13, I17, I19, I31. Oi, /oi I31 – i?inoia /enei,
aeieacae a 1750 aiaeo E. Yeea?. A 1876 aiaeo o?aioeocneee iaoaiaoee
Yaeoa?ae Ethea

onoaiiaee, /oi /enei

3)I127=170141183460469231731687303715884105727

– i?inoia. A 1883 a. Naeueneee nayuaiiee Ia?ineie aoaa?iee E.I.Ia?aooei
aac anyeeo au/eneeoaeueiuo i?eai?ia aeieacae, /oi /enei
I61=2305843009213693951 yaeyaony i?inoui. Iicaeiaa auei onoaiiaeaii,
/oi /enea I89 e I107 – i?inoua. Eniieueciaaiea YAI iicaieeei a 1952-1964
aiaeao aeieacaoue, /oi /enea I521, I607, I1279, I2203, I2281, I3217,
I4253, I4423, I2689, I9941, I11213 – i?inoua. E ianoiyuaio a?aiaie
ecaanoii oaea aieaa 30 i?inouo /enae Ia?naiia, iaeii ec eioi?uo I216091
eiaao 65050 oeeo?. Aieueoie eioa?an e i?inoui /eneai Ia?naiia aucaai eo
oaniie naycueth n niaa?oaiiuie /eneaie.

Iaoo?aeueiia /enei ? iacuaaaony niaa?oaiiui, anee iii ?aaii noiia anao
naieo aeaeeoaeae e?iia ?.

Aaeeeae aeieacae, /oi anee ? e 2?-1 – i?inoua /enea, oi /enei
4)??=2?-1(2?-1)=2?-1I? yaeyaony niaa?oaiiui.

Aeaenoaeoaeueii, aeaeeoaeyie oaeiai /enea, aeeth/ay naii yoi /enei,
yaeythony 5)1,2, … ,2?-1,I?,2I?, … ,2?-1I? .

Eo noiia Sp=(1+2+ … +2?-1)(I?+1)=(2?-1) . 2?=2 . 2?-1 I?. Au/eoay ec S
naii /enei ?? , oaaaeaeaainy, /oi noiia anao aeaeeoaeae /enea ?? ?aaia
yoiio /eneo, neaaeiaaoaeueii ?? – niaa?oaiiia /enei.

*enea ?2=6 e ?3=28 auee ecaanoiu au? ieoaai?aeoeai. *enea ?5=496 e
?7=8128 iaoae Aaeeeae. Eniieuecoy ae?oaea i?inoua /enea Ia?naiia e
oi?ioeo 4, iaoiaeei neaaeothuea niaa?oaiiua /enea:

6)?13=33550336, ?17=8589869056, ?19=137438691328,
?31=2305843008139952128.

Aeey anao inoaeueiuo /enae Ia?naiia /enea ?? eiatho i/aiue iiiai oeeo?.

Aei neo ii? inoa?ony caaaaeeie, eae Ia?naii niia auneacaoue i?aaeeueiia
ooaa?aeaeaiea, /oi /enea ?17, ?19, ?31 yaeythony niaa?oaiiuie. Iicaeiaa
auei iaia?oaeaii, /oi ii/oe ca noi eao aei Ia?naiia /enea ?17, ?19 iaoae
eoaeueyineee iaoaiaoee Eaoaeueaee – i?ioanni? oieaa?neoaoia Oei?aioeee e
Aieiiuee. N/eoaeinue, /oi aiaeanoaaiiia i?iaeaeaiea i?aaeneacaei naiei
eca?aiieeai i?aaeeueiua cia/aiey yoeo niaa?oaiiuo /enae. Anee o/anoue,
/oi au? ieoaai?aeoeu n/eoaee ia?aia niaa?oaiiia /enei 6 neiaieii aeooe,
/oi aoi?ia niaa?oaiiia /enei 28 niioaaonoaiaaei /eneo /eaiia iiiaeo
o/?iuo iauanoa, /oi aeaaea a aeaaiaaeoeaoii aaea oea?eiaue o/eea: aeey
nianaiey aeooe aeinoaoi/ii eco/aoue niaa?oaiiua /enea e oiio, eoi
iaeae?o iiaia aiaeanoaaiiia niaa?oaiiia /enei, oaioiaaii aa/iia
aeaaeainoai, oi noaiiaeony iiiyoiui eneeth/eoaeueiue eioa?an e yoei
/eneai.

Iaeiaei e n iaoaiaoe/aneie oi/ee c?aiey /?oiua niaa?oaiiua /enea
ii-naiaio oieeaeueiu. Ana iie – o?aoaieueiua. Noiia aaee/ei, ia?aoiuo
anai aeeeeoaeyi /enea, aeeth/ay naii /enei, anaaaea ?aaia aeaoi. Inoaoie
io aeaeaiey niaa?oaiiiai /enea, e?iia 6, ia 9 ?aaai 1. A aeaie/iie
nenoaia niaa?oaiiia /enei ?? ia/eiaaony ? aaeeieoeaie, iioii neaaeotho
?-1 ioeae. Iai?eia?:

7)?2=110, ?3=11100, ?5 =111110000, ?7 =1111111000000 e o.ae.

Iineaaeiyy oeeo?a /?oiiai niaa?oaiiiai /enea eee 6, eee 8, i?e/?i, anee
8, oi ae i?aaeoanoaoao 2.

Eaiia?ae Yeea? aeieacae, /oi ana /?oiua niaa?oaiiua /enea eiatho aeae
2?-1 . I?, aaea I?-i?inoia /enei Ia?naiia. Iaeiaei aei neo ii? ia
iaeaeaii ie iaeiiai ia/?oiiai niaa?oaiiiai /enea. Auneacaii
i?aaeiieiaeaiea(A?aeai Oaeoa?iai,NOA), /oi anee oaeia /enei nouanoaoao,
oi iii aeieaeii eiaoue ia iaiaa 36 ciaeia.

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