Kev txiav txim los ntawm qhov muaj koob meej ntawm qhov kev thov "Fermat's theorem - luv luv pov thawj", qhov teeb meem lej no yog qhov txaus siab rau ntau tus. Qhov kev xav no yog thawj zaug tau hais los ntawm Pierre de Fermat hauv 1637 ntawm ntug ntawm daim ntawv theej ntawm Arithmetic, qhov uas nws tau thov tias nws muaj cov kev daws teeb meem loj heev kom haum rau ntawm ntug.
Thawj cov pov thawj ua tiav tau luam tawm xyoo 1995 - nws yog cov pov thawj tiav ntawm Fermat's Theorem los ntawm Andrew Wiles. Nws tau raug piav raws li "kev vam meej" thiab coj Wiles kom tau txais Abel nqi zog hauv 2016. Txawm hais tias tau piav qhia luv luv, cov pov thawj ntawm Fermat's theorem kuj tau ua pov thawj ntau ntawm modularity theorem thiab qhib txoj hauv kev tshiab rau ntau lwm yam teeb meem thiab cov txheej txheem zoo rau kev txhawb nqa modularity. Cov kev ua tiav no tau nce qib lej 100 xyoo rau yav tom ntej. Cov pov thawj ntawm Fermat lub me theorem niaj hnub no tsis yogyog ib yam dab tsi tawm ntawm qhov dog dig.
Qhov teeb meem tsis daws teeb meem tau txhawb txoj kev loj hlob ntawm algebraic tus lej txoj kev xav nyob rau hauv lub xyoo pua 19th thiab kev tshawb nrhiav cov pov thawj ntawm modularity theorem nyob rau hauv lub xyoo pua 20th. Qhov no yog ib qho tseem ceeb tshaj plaws theorems nyob rau hauv keeb kwm ntawm kev ua lej, thiab kom txog rau thaum tag nrho cov ntaub ntawv pov thawj ntawm Fermat's Last Theorem, nws yog nyob rau hauv Guinness Phau Ntawv Cov Ntaub Ntawv raws li "qhov teeb meem kev ua lej nyuaj tshaj plaws", ib qho ntawm cov yam ntxwv uas yog qhov ntawd. nws muaj cov ntaub ntawv pov thawj ntau tshaj plaws.
keeb kwm keeb kwm
Pythagorean equation x2 + y2=z2 muaj tus lej tsis kawg integer cov ntsiab lus rau x, y thiab z. Cov kev daws teeb meem no yog hu ua Pythagorean trinities. Nyob ib ncig ntawm 1637, Fermat tau sau rau ntawm ntug ntawm phau ntawv tias ntau qhov sib npaug a + b =ctsis muaj cov kev daws teeb meem hauv cov lej yog n yog tus lej ntau dua 2. Txawm hais tias Fermat nws tus kheej tau thov kom muaj kev daws teeb meem rau nws qhov teeb meem, nws tsis tau tso ib qho lus qhia txog nws cov pov thawj. Cov pov thawj tseem ceeb ntawm Fermat's theorem, thov los ntawm nws tus tsim, yog qhov nws khav theeb. Phau ntawv ntawm tus lej Fabkis txoj lej tau tshawb pom 30 xyoo tom qab nws tuag. Qhov kev sib npaug no, hu ua Fermat's Last Theorem, tseem tsis tau daws nyob rau hauv lej rau peb thiab ib nrab ib-paus xyoo.
Theorem nws thiaj li dhau los ua ib qho teeb meem tseem ceeb tshaj plaws uas tsis tau daws teeb meem hauv lej. Kev sim ua pov thawj qhov no ua rau muaj kev txhim kho tseem ceeb ntawm tus lej txoj kev xav, thiab nrog nqe luslub sij hawm, Fermat lub xeem theorem tau paub tias yog ib qho teeb meem tsis daws teeb meem hauv lej.
Ab Keeb Kwm Ntawm Pov Thawj
Yog n=4, raws li tau ua pov thawj los ntawm Fermat nws tus kheej, nws txaus los ua pov thawj lub theorem rau indices n uas yog cov lej tseem ceeb. Ob xyoo tom ntej no (1637-1839) qhov kev xav tau tsuas yog ua pov thawj rau cov thawj 3, 5 thiab 7, txawm hais tias Sophie Germain tau hloov kho thiab ua pov thawj txoj hauv kev uas siv rau tag nrho cov chav kawm ntawm primes. Nyob rau hauv nruab nrab-19th caug xyoo, Ernst Kummer tau txuas ntxiv qhov no thiab ua pov thawj lub theorem rau tag nrho cov primes tsis tu ncua, uas tsis xwm yeem primes tau txheeb xyuas tus kheej. Raws li Kummer txoj haujlwm thiab siv cov kev tshawb fawb hauv computer, lwm cov lej muaj peev xwm txuas ntxiv qhov kev daws teeb meem ntawm lub theorem, nrog rau lub hom phiaj los npog tag nrho cov ntsiab lus tseem ceeb txog plaub lab, tab sis cov pov thawj rau tag nrho cov exponents tseem tsis tau muaj (lub ntsiab lus tias cov lej lej. feem ntau suav hais tias yog qhov kev daws teeb meem ntawm qhov theorem ua tsis tau, nyuaj heev, lossis tsis tuaj yeem ua tiav nrog kev paub tam sim no).
Kev ua haujlwm ntawm Shimura thiab Taniyama
Nyob rau xyoo 1955, tus lej Japanese Goro Shimura thiab Yutaka Taniyama xav tias muaj kev sib txuas ntawm elliptic curves thiab modular forms, ob ceg sib txawv ntawm kev ua lej. Paub thaum lub sij hawm raws li Taniyama-Shimura-Weyl conjecture thiab (thaum kawg) raws li lub modularity theorem, nws muaj nyob rau ntawm nws tus kheej, tsis muaj kev sib txuas rau Fermat lub xeem theorem. Nws tus kheej tau dav suav hais tias yog ib qho tseem ceeb ntawm kev ua lej, tab sis nws tau raug suav hais tias (xws li Fermat's theorem) tsis tuaj yeem ua pov thawj. Thaum ntawdTib lub sijhawm, cov pov thawj ntawm Fermat's Last Theorem (los ntawm kev faib thiab siv cov qauv lej nyuaj) tau ua tiav ib nrab xyoo tom qab.
Hauv xyoo 1984, Gerhard Frey tau pom muaj kev sib raug zoo ntawm ob qhov teeb meem yav dhau los thiab tsis tau daws teeb meem. Kev lees paub tiav tias ob txoj kev xav tau zoo sib xws tau luam tawm xyoo 1986 los ntawm Ken Ribet, uas ua raws li cov pov thawj ib nrab los ntawm Jean-Pierre Serra, uas tau ua pov thawj tag nrho tab sis ib feem, hu ua "epsilon hypothesis". Yooj yim muab tso, cov haujlwm no los ntawm Frey, Serra, thiab Ribe tau pom tias yog qhov kev xav ntawm modularity tuaj yeem ua pov thawj, yam tsawg kawg rau cov chav kawm semistable ntawm elliptic curves, ces cov pov thawj ntawm Fermat lub theorem kawg yuav pom sai dua lossis tom qab ntawd. Txhua qhov kev daws teeb meem uas tuaj yeem cuam tshuam Fermat lub theorem kawg kuj tuaj yeem siv los cuam tshuam qhov kev xav ntawm modularity. Yog li ntawd, yog hais tias lub modularity theorem tig los ua qhov tseeb, ces los ntawm txhais tau tias tsis muaj ib qho kev daws teeb meem uas cuam tshuam Fermat lub xeem theorem, uas txhais tau hais tias nws yuav tsum tau ua pov thawj sai sai.
Txawm hais tias ob qho tib si theorems yog teeb meem nyuaj hauv kev ua lej, suav tias tsis tuaj yeem daws tau, kev ua haujlwm ntawm ob tus neeg Nyij Pooj yog thawj qhov kev tawm tswv yim ntawm Fermat lub theorem kawg tuaj yeem txuas ntxiv thiab ua pov thawj rau txhua tus lej, tsis yog qee qhov. Ib qho tseem ceeb rau cov kws tshawb fawb uas tau xaiv lub ntsiab lus ntawm kev kawm yog qhov tseeb hais tias, nyob rau hauv sib piv rau Fermat lub xeem theorem, lub modularity theorem yog lub ntsiab active cheeb tsam ntawm kev tshawb fawb, uas.cov pov thawj tau tsim, thiab tsis yog keeb kwm oddity xwb, yog li lub sijhawm siv rau nws txoj haujlwm tuaj yeem raug lees paub los ntawm cov kws tshaj lij pom. Txawm li cas los xij, qhov kev pom zoo dav dav yog tias kev daws qhov kev xav ntawm Taniyama-Shimura tau ua pov thawj tias tsis tsim nyog.
Farm's Last Theorem: Wiles' pov thawj
Tom qab kawm tias Ribet tau ua pov thawj Frey txoj kev xav yog lawm, Askiv tus lej lej Andrew Wiles, uas tau txaus siab rau Fermat's Last Theorem txij thaum yau thiab muaj kev paub ua haujlwm nrog elliptic nkhaus thiab thaj chaw nyob ib sab, txiav txim siab sim ua pov thawj Taniyama-Shimura Kev xav raws li ib txoj hauv kev los ua pov thawj Fermat's Last Theorem. Xyoo 1993, rau xyoo tom qab tshaj tawm nws lub hom phiaj, thaum ua haujlwm zais cia ntawm qhov teeb meem ntawm kev daws qhov theorem, Wiles tau tswj hwm los ua pov thawj muaj feem cuam tshuam, uas yuav pab nws ua pov thawj Fermat lub ntsiab lus kawg. Wiles 'cov ntaub ntawv yog qhov loj thiab muaj peev xwm.
Ib qhov tsis txaus ntseeg tau pom nyob rau hauv ib feem ntawm nws daim ntawv qub thaum lub sijhawm tshuaj xyuas thiab xav tau lwm xyoo ntawm kev koom tes nrog Richard Taylor los koom ua ke daws qhov kev xav. Yog li ntawd, Wiles 'cov pov thawj zaum kawg ntawm Fermat's Last Theorem tsis ntev los no. Xyoo 1995, nws tau luam tawm ntawm qhov me me dua li Wiles txoj haujlwm ua lej yav dhau los, qhia tias nws tsis tau yuam kev hauv nws cov lus xaus dhau los txog qhov muaj peev xwm ua pov thawj lub theorem. Wiles 'kev ua tau zoo tau tshaj tawm thoob plaws hauv cov xov xwm nrov thiab nrov hauv phau ntawv thiab TV programs. Qhov seem ntawm Taniyama-Shimura-Weil kev xav, uas tam sim no tau raug pov thawj thiablub npe hu ua modularity theorem, tom qab ntawd tau ua pov thawj los ntawm lwm tus lej uas tau tsim los ntawm Wiles txoj haujlwm ntawm 1996 thiab 2001. Rau nws qhov kev ua tiav, Wiles tau txais txiaj ntsig thiab tau txais ntau yam khoom plig, suav nrog 2016 Abel nqi zog.
Wiles 'cov pov thawj ntawm Fermat's theorem kawg yog ib rooj plaub tshwj xeeb ntawm kev daws qhov modularity theorem rau elliptic curves. Txawm li cas los xij, qhov no yog cov ntaub ntawv nto moo tshaj plaws ntawm xws li kev ua lej loj. Nrog rau kev daws Ribe's theorem, tus kws lej Askiv kuj tau txais cov pov thawj ntawm Fermat lub theorem kawg. Fermat's Last Theorem thiab Modularity Theorem yuav luag universally suav hais tias yog unprovable los ntawm niaj hnub mathematicians, tab sis Andrew Wiles muaj peev xwm ua pov thawj rau lub ntiaj teb kev tshawb fawb tias txawm pundits yuav ua tsis ncaj ncees lawm.
Wyles thawj zaug tshaj tawm nws qhov kev tshawb pom rau hnub Wednesday 23 Lub Rau Hli 1993 ntawm Cambridge cov lus qhuab qhia hu ua "Modular Forms, Elliptic Curves thiab Galois Representations". Txawm li cas los xij, thaum lub Cuaj Hlis 1993, nws tau pom tias nws cov kev suav suav muaj qhov yuam kev. Ib xyoos tom qab, thaum lub Cuaj Hlis 19, 1994, hauv qhov nws yuav hu ua "lub sijhawm tseem ceeb tshaj plaws ntawm nws lub neej ua haujlwm," Wiles tau ua rau muaj kev tshwm sim uas tso cai rau nws los kho qhov kev daws teeb meem mus txog qhov uas nws tuaj yeem txaus siab rau kev ua lej. community.
piav qhia ua haujlwm
Pov thawj ntawm Fermat's Theorem los ntawm Andrew Wiles siv ntau txoj hauv kev los ntawm algebraic geometry thiab tus lej txoj kev xav thiab muaj ntau qhov cuam tshuam hauv cov nothaj chaw ntawm lej. Nws kuj tseem siv cov qauv tsim ntawm niaj hnub algebraic geometry, xws li qeb ntawm schemes thiab Iwasawa txoj kev xav, nrog rau lwm txoj hauv kev ntawm xyoo pua 20th uas tsis muaj rau Pierre de Fermat.
Ob tsab ntawv uas muaj cov pov thawj yog 129 nplooj ntawv ntev thiab tau sau nyob rau hauv xya xyoo. John Coates tau piav qhia qhov kev tshawb pom no yog ib qho kev ua tiav zoo tshaj plaws ntawm cov lej thev naus laus zis, thiab John Conway tau hu nws qhov kev ua lej loj ntawm xyoo pua 20th. Wiles, txhawm rau ua pov thawj Fermat lub xeem theorem los ntawm kev ua pov thawj qhov kev xav ntawm modularity rau cov ntaub ntawv tshwj xeeb ntawm semistable elliptic curves, tsim cov txheej txheem muaj zog rau kev txhawb nqa modularity thiab qhib txoj hauv kev tshiab rau ntau yam teeb meem. Rau kev daws Fermat lub xeem theorem, nws tau knighted thiab tau txais lwm yam khoom plig. Thaum nws paub tias Wiles tau yeej Abel nqi zog, Norwegian Academy of Sciences tau piav qhia nws qhov kev ua tiav yog "ib qho pov thawj zoo siab thiab theem pib ntawm Fermat lub theorem kawg."
Zoo li cas
Ib tug ntawm cov neeg uas tshuaj xyuas Wiles 'cov ntawv sau qub nrog rau kev daws rau lub theorem yog Nick Katz. Hauv chav kawm ntawm nws qhov kev tshuaj xyuas, nws tau nug Briton ntau cov lus nug meej uas ua rau Wiles lees paub tias nws txoj haujlwm muaj qhov sib txawv. Nyob rau hauv ib qho tseem ceeb ntawm cov ntaub ntawv pov thawj, ib qho kev ua yuam kev tau muab kev kwv yees rau qhov kev txiav txim ntawm ib pab pawg neeg: Euler system siv los txuas ntxiv txoj kev Kolyvagin thiab Flach tsis tiav. Qhov yuam kev, txawm li cas los xij, tsis ua rau nws txoj haujlwm tsis muaj txiaj ntsig - txhua qhov ntawm Wiles txoj haujlwm tseem ceeb heev thiab tsim kho tshiab hauv nws tus kheej, zoo li ntau yam.kev txhim kho thiab cov txheej txheem uas nws tsim nyob rau hauv chav kawm ntawm nws txoj haujlwm thiab uas cuam tshuam tsuas yog ib feem ntawm cov ntawv sau. Txawm li cas los xij, cov haujlwm qub no, luam tawm xyoo 1993, tsis muaj pov thawj ntawm Fermat's Last Theorem.
Wyles siv sijhawm ze li ib xyoos sim rov tshawb nrhiav qhov kev daws teeb meem rau txoj kev xav, ua ntej ib leeg thiab tom qab ntawd koom tes nrog nws tus tub ntxhais kawm qub Richard Taylor, tab sis txhua yam zoo li tsis muaj dab tsi. Thaum kawg ntawm xyoo 1993, cov lus xaiv tau tshaj tawm tias Wiles cov pov thawj tau ua tsis tiav hauv kev sim, tab sis qhov kev ua tsis tiav loj npaum li cas tsis paub. Cov kws lej tau pib ua rau Wiles nthuav qhia cov ntsiab lus ntawm nws txoj haujlwm, txawm tias nws tau ua tiav lossis tsis ua, kom lub zej zog dav ntawm cov lej tuaj yeem tshawb nrhiav thiab siv txhua yam uas nws muaj peev xwm ua tiav. Hloov chaw sai sai kho nws qhov yuam kev, Wiles tsuas yog nrhiav pom qhov nyuaj ntxiv hauv cov pov thawj ntawm Fermat's Last Theorem, thiab thaum kawg pom tias nws nyuaj npaum li cas.
Wyles hais tias thaum sawv ntxov lub Cuaj Hlis 19, 1994, nws tau tawm tsam thiab tso tseg, thiab yuav luag tawm haujlwm ua tsis tiav. Nws tau npaj los tshaj tawm nws txoj haujlwm tsis tiav kom lwm tus tuaj yeem tsim kho thiab nrhiav qhov twg nws tsis raug. Tus kws lej Askiv txiav txim siab muab sijhawm kawg rau nws tus kheej thiab tshuaj xyuas lub theorem rau lub sijhawm kawg los sim nkag siab txog cov laj thawj tseem ceeb vim li cas nws txoj hauv kev tsis ua haujlwm, thaum nws mam li nco dheev pom tias txoj hauv kev Kolyvagin-Flac yuav tsis ua haujlwm txog thaum nws.tseem yuav suav nrog Iwasawa txoj kev xav hauv cov txheej txheem pov thawj, ua kom nws ua haujlwm.
Lub Kaum Hli 6, Wiles nug peb tus npoj yaig (xws li F altins) los tshuaj xyuas nws txoj haujlwm tshiab, thiab thaum Lub Kaum Hli 24, 1994, nws tau xa ob daim ntawv sau - "Modular elliptic curves thiab Fermat's theorem kawg" thiab "Theoretical properties of the lub nplhaib ntawm qee qhov Hecke algebras ", qhov thib ob uas Wiles tau sau nrog Taylor thiab ua pov thawj tias qee qhov xwm txheej tau ntsib los ua kom pom tseeb cov kauj ruam kho nyob rau hauv tsab xov xwm tseem ceeb.
Ob daim ntawv no tau raug tshuaj xyuas thiab thaum kawg tau luam tawm ua cov ntawv sau tag nrho hauv lub Tsib Hlis 1995 Annals of Mathematics. Andrew cov kev suav tshiab tau raug tshuaj xyuas dav dav thiab thaum kawg tau txais los ntawm cov zej zog kev tshawb fawb. Nyob rau hauv cov ntaub ntawv no, lub modularity theorem rau semistable elliptic curves tau tsim - lub kauj ruam kawg ntawm kev ua pov thawj Fermat's Last Theorem, 358 xyoo tom qab nws tau tsim.
History of the Great problem
Kev daws qhov kev xav no tau suav tias yog qhov teeb meem loj tshaj plaws hauv kev ua lej rau ntau pua xyoo. Xyoo 1816 thiab xyoo 1850 Fabkis Academy ntawm Kev Tshawb Fawb tau muab khoom plig rau cov pov thawj dav dav ntawm Fermat's Last Theorem. Xyoo 1857, Academy tau muab 3,000 francs thiab ib qho khoom plig kub rau Kummer rau nws txoj kev tshawb fawb ntawm cov lej zoo, txawm tias nws tsis tau thov rau qhov khoom plig. Lwm qhov khoom plig tau muab rau nws xyoo 1883 los ntawm Brussels Academy.
Wolfskell nqi zog
Hauv xyoo 1908, German tus kws ua lag luam thiab tus kws ua lej amateur Paul Wolfskel bequeathed 100,000 kub (ntau npaum li cas rau lub sijhawm ntawd)Academy of Sciences ntawm Göttingen, kom cov nyiaj no dhau los ua khoom plig rau cov pov thawj tiav ntawm Fermat lub theorem kawg. Thaum Lub Rau Hli 27, 1908, Academy tau tshaj tawm cuaj qhov khoom plig. Ntawm lwm yam, cov kev cai no xav tau cov ntaub ntawv pov thawj los luam tawm hauv phau ntawv xov xwm sib tham. Qhov khoom plig yuav tsum tau muab tsub tsuas yog ob xyoos tom qab luam tawm. Qhov kev sib tw yuav tas rau lub Cuaj Hlis 13, 2007 - txog ib puas xyoo tom qab nws pib. Thaum Lub Rau Hli 27, 1997, Wiles tau txais Wolfschel cov nqi zog thiab tom qab ntawd ntxiv $ 50,000. Thaum Lub Peb Hlis 2016, nws tau txais € 600,000 los ntawm tsoomfwv Norwegian uas yog ib feem ntawm Abel nqi zog rau "ib qho pov thawj zoo kawg nkaus ntawm Fermat lub theorem kawg nrog kev pab los ntawm cov kev xav ntawm modularity rau semistable elliptic curves, qhib lub sijhawm tshiab ntawm cov lej kev xav." Nws yog lub ntiaj teb kev yeej ntawm cov neeg Askiv uas txo hwj chim.
Ua ntej Wiles 'cov pov thawj, Fermat's theorem, raws li tau hais dhau los, tau suav tias yog qhov tsis tuaj yeem daws tau ntau pua xyoo. Ntau txhiab cov pov thawj tsis raug nyob rau ntau lub sijhawm tau nthuav tawm rau pawg neeg Wolfskell, suav nrog kwv yees li 10 feet (3 meters) ntawm kev sau ntawv. Tsuas yog thawj xyoo ntawm qhov muaj txiaj ntsig (1907-1908) 621 daim ntawv thov raug xa mus thov kom daws qhov kev xav, txawm hais tias los ntawm xyoo 1970 lawv cov lej tau txo mus txog 3-4 daim ntawv thov ib hlis. Raws li F. Schlichting, Wolfschel tus kws tshuaj xyuas, feem ntau ntawm cov pov thawj tau ua raws li cov txheej txheem theem pib qhia hauv cov tsev kawm ntawv thiab feem ntau tau nthuav tawm tias "cov neeg muaj keeb kwm txuj ci tab sis tsis ua tiav txoj haujlwm". Raws li tus historian ntawm lej Howard Aves, kawgFermat's theorem tau teeb tsa ib hom ntaub ntawv - qhov no yog lub theorem uas muaj cov pov thawj loj tshaj plaws.
Farm's laurels mus rau Japanese
Raws li tau hais ua ntej, nyob ib puag ncig xyoo 1955, tus lej Japanese Goro Shimura thiab Yutaka Taniyama tau pom muaj kev sib txuas ntawm ob ceg sib txawv ntawm kev ua lej - elliptic curves thiab modular forms. Qhov tshwm sim modularity theorem (tom qab ntawd hu ua Taniyama-Shimura conjecture) hais tias txhua qhov elliptic nkhaus yog modular, txhais tau hais tias nws tuaj yeem cuam tshuam nrog ib daim ntawv tshwj xeeb.
Qhov kev xav tau raug tso tseg thaum xub thawj tsis zoo li qhov tsis zoo lossis qhov kev xav tsis zoo, tab sis raug coj mus rau qhov hnyav dua thaum tus lej theorist André Weil pom cov pov thawj los txhawb cov lus Nyij Pooj. Yog li ntawd, qhov kev xav tau feem ntau raug xa mus rau Taniyama-Shimura-Weil hypothesis. Nws tau los ua ib feem ntawm txoj haujlwm Langlands, uas yog cov npe ntawm cov kev xav tseem ceeb uas yuav tsum tau ua pov thawj yav tom ntej.
Txawm tias tom qab kev tshuaj xyuas hnyav, qhov kev xav tau raug lees paub los ntawm cov kws lej niaj hnub no tias nyuaj heev, lossis tej zaum tsis tuaj yeem ua pov thawj. Tam sim no qhov kev xav tshwj xeeb no tab tom tos nws Andrew Wiles, uas tuaj yeem ua rau lub ntiaj teb xav tsis thoob nrog nws cov kev daws teeb meem.
Fermat's Theorem: Perelman's proof
Txawm hais tias muaj lus dab neeg nrov, tus lej Lavxias teb sab Grigory Perelman, rau tag nrho nws cov neeg txawj ntse, tsis muaj dab tsi cuam tshuam nrog Fermat lub theorem. Uas, txawm li cas los, nyob rau hauv tsis muaj txoj kev detracts los ntawm nws.ntau qhov kev pab cuam rau lub zej zog scientific.
