Yog kuj prime. foundation

Mathematics phau ntawv tej zaum kuj tsis yooj yim rau to taub. Qhuav thiab meej lus tus sau phau ntawv tsis yog ib qho yooj yim to taub. Thiab muaj yeej ib txwm ua ke cov ncauj lus, vzaimovytekayuschie. Rau txoj kev loj hlob ntawm lub ntsiab nws yog tsim nyog los tsa ib tug xov tooj ntawm yav tas los thiab tej zaum kuj tig los ntawm tag nrho cov phau ntawv nyeem. Nyuab? Yog. Wb twv circumvent cov teeb meem thiab ua kom nrhiav tau cov npe no yog tsis heev lub txheem mus kom ze. Peb ua ib tug zoo ntawm excursion mus rau hauv lub teb chaws xov tooj. Lus Txhais, txawm li cas los, peb tseem nyob li qub, vim hais tias cov kev cai ntawm kev kawm txog zauv yuav tsis tau undone. Yog li ntawd, kuj prime xov tooj - tus xov tooj ntawm tej yam ntuj tso, nrog ib tug ntau divisor sib npaug zos rau ib tug. Yog hais tias to taub? Nws yog.

Rau ib tug ntau nraaj piv txwv li, cia wb mus rau hauv lub xov tooj 6 thiab 13. Thiab ces, thiab ntau - yog divisible los ntawm ib tug (kuj prime). Tab sis cov zauv 12 thiab 14 - xws li tsis tau yuav, vim hais tias lub caij nplooj zeeg tsis yog tsuas yog 1, tab sis kuj nyob rau hauv lub 2 nram qab no tus xov tooj - 21 thiab 47 kuj tsis haum rau hauv qeb ntawm "kuj prime": lawv yuav tsum tau muab faib tsis tau tsuas yog 1, tab sis kuj 7.

Txhais kuj prime xov tooj raws li (ib tug, y) = 1.

Peb yuav hais txawm ntau tsuas: ntau divisor (siab tshaj plaws) yog sib npaug zos rau ib tug.
Vim li cas peb muaj tej kev kawm? Vim li cas txaus.

Nrog prime xov tooj muaj nyob rau hauv ib co encryption system. Cov neeg uas ua hauj lwm nrog cov Hill cipher los yog Caesar rewriting systems, to taub hais tias tsis muaj txoj kev paub no - nyob qhov twg. Yog hais tias koj twb hnov txog ib tug random tooj generator, nws yog tsis zoo li mus twv tau tsis kam lees: kuj prime tus xov tooj yog siv thiab muaj.

Tam sim no wb tham txog yuav ua li cas kom tau cov xov tooj. Tus nab npawb ntawm tej yam yooj yim, raws li koj paub, tej zaum yuav muaj tsuas yog ob tug divisors: lawv faib los ntawm lawv tus kheej thiab los ntawm ib tug. Hais tias, 11, 7, 5, 3 - lub xov tooj ntawm tej yam yooj yim, tab sis 9 - tsis muaj, nws yog twb tus xov tooj yog divisible thiab 9, thiab 3, thiab 1.

Thiab yog hais tias ib tug - ib tug prime xov tooj, thaum - nyob rau hauv lub teeb {1, 2, ... thiab - 1}, ces guaranteed (a, y) = 1, los yog nrog prime xov tooj - ib tug thiab y.

Nws yog ib, es, twb tsis muaj piav thiab repetition los yog ntsiab lus dab tsi tau hais tias.

Tau primes tejzaum sab cib ntawm Eratosthenes, tab sis rau cov impressive xov tooj (billions, piv txwv li), cov qauv no yog dhau lawm ntev, tab sis, tsis zoo li tus super-mis nyuj tov, uas tej zaum kuj ua yuam kev, ntau txhim khu kev qha.

Koj muaj peev xwm ua hauj lwm los ntawm xaiv los ntawm> a. Ua li no, nws yog xaiv li ntawd tus xov tooj ntawm rau thiab tsis muab faib. Rau lub hom phiaj no, ib tug prime xov tooj yog multiplied los ntawm ib tug tej yam ntuj tso xov tooj thiab yog ntxiv (los yog, hloov, rho tawm) tus nqi (piv txwv li, p), uas yog tsawg zoo:

y = p + k thiab

Yog hais tias, piv txwv li, ib tug = 71, p = 3, q = 10, ces, raws li, yuav muaj sib npaug zos rau 713. Lwm tau xaiv, nrog degrees.

Tiabsis cov xov tooj raws li txwv mus kuj prime, thiab cov qhia, thiab 1, thiab lwm yam xov tooj (kuj tsis muaj seem).

Nyob rau hauv lwm yam lus, lub ntuj tus xov tooj (tshwj tsis yog rau ib qho) muab faib ua feem thiab yooj yim.

Prime numbers - tus xov tooj ntawm tej yam ntuj tso, uas tsis yog-tsis tseem ceeb (txawv los ntawm cov xov tooj thiab cov chav nyob) dividers. Tshwj xeeb tshaj yog ib qho tseem ceeb yog lawv lub luag hauj lwm nyob rau hauv hnub no txoj kev niaj hnub, fast-paced cryptography, ua tsaug rau uas lub hom phiaj ntawm tus xov tooj, yav tas los xav heev paub daws teeb kev qhuab qhia, tau ua thiaj li nyob rau hauv coob: cov ntaub ntawv tiv thaiv algorithms yog lossi tau zoo tuaj.

Qhov loj tshaj plaws prime tooj pom ib tug kws kho mob-ophthalmologist Martin Novak, uas koom nyob rau hauv peb tes num GIMPS (distributive xam) ua ke nrog rau lwm cov enthusiasts, uas suav txog 15 txhiab. Nyob rau hauv cov lus teb coj rau ntev xyoo. ob thiab ib tug ib nrab teb computers nyob rau hauv lub qhov muag lub tsev kho mob Novak tau muab kev koom tes. Cov tshwm sim ntawm Titanic ua hauj lwm thiab kev rau siab yog lub xov tooj 225964951-1, sau rau ib tug 7,816,230-nyob rau hauv decimals. Los ntawm txoj kev, cov ntaub ntawv ntawm cov loj tshaj plaws pes tsawg tus twb xa rau lub hlis ua ntej qhib. Thiab muaj tej yam tshwm sim nyob rau hauv qis dua ib nrab.

Peb ntse uas xav kom koj hu rau ib tug xov tooj, qhov chaw uas lub duration ntawm lub decimal "dhia" kaum-millionth cim, muaj ib lub caij tau tsis tsuas thoob ntiaj teb koob meej tab sis kuj $ 100 000. Los ntawm txoj kev, rau cov xov tooj tau kov yeej millionth milestone qhab nia Nayan Hayratval tau txais ib tug tsawg dua tus nqi (50 000 dollars).