Transcription of An Overview of the Proof of Fermat’s Last Theorem
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An Overview of the Proof of Fermat’s Last Theorem
AnOverviewoftheProofofFermat >2,wehaveFLT(n):an+bn=cna,b,c2Z =)abc= ,whoprovedFLT(4) (3) (d)=)FLT(n)wheneverd|n, 5isprime,anda,b,c2Z,thenap+bp+cp=0=)abc= ortsofinnumer-ablemathematicianswhohavew orkedoverthelastcentury(andmore!)todevel oparichandpowerfularithmetictheoryofelli pticcurves,modularforms, sLastTheoremandtosupplythefinalcrucialin gredientsoftheproof:GerhartFrey(1985),wh ofirstsuggestedthattheexistenceofasolu-t ionoftheFermatequationmightcontradictthe ModularityConjectureofTaniyama,Shimura,a ndWeil;Jean-PierreSerre(1985-6),whoformu latedand( )testednumericallyapreciseconjectureabou tmodularformsandgaloisrep-resentationsmo dpandwhoshowedhowasmallpieceofthisconjec ture theso-calledepsilonconjecture togetherwiththeModularityConjecturewould implyFermat sLastTheorem.
To prove the theorem we follow the program outlined by Serre in [16]. Fix a prime p 5 and suppose a,b,c 2 Z satisfy ap + bp + cp = 0 but abc 6= 0. The triple (ap,bp,cp) is what Gerhard Frey has called a “remark- able” triple of integers, so remarkable in fact, that we suspect it …
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