The Solving Of Fermat's Last Theorem
The Solving of Fermat’s Last TheoremKarl RubinEdward and Vivian Thorp Professor of MathematicsMarch 20, 2007Physical Sciences Breakfast LectureKarl Rubin (UC Irvine)Fermat’s Last Theorem1PS Breakfast, March 20071 / 37
Pythagorean TheoremA2 B 2 C 232 42 52C52 122 132A82 152 172392 802 892···BKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 20072 / 37
Plympton 322Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 20073 / 37
Plympton 270090Karl Rubin (UC 92893229106Fermat’s Last Theoremangle45.2 45.7 46.2 46.7 47.9 48.5 49.7 50.2 51.3 52.6 53.1 55.0 56.1 56.7 58.1 PS Breakfast, March 20074 / 37
DiophantusKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 20075 / 37
FermatKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 20076 / 37
FermatKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 20077 / 37
Fermat’s Last Theorem“It is impossible to separate a cube into two cubes,a3 b3 c 3 has no whole number solutionsor a fourth power into two fourth powers,a4 b4 c 4 has no whole number solutionsor in general any power greater than the second into two likepowers.”Fermat’s Last TheoremIf n 2 then an bn c n has no whole number solutions.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 20078 / 37
Fermat’s Last Theorem“I have discovered a trulymarvelous proof of this,which this margin is notlarge enough to contain.”Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 20079 / 37
Fermat’s Last TheoremKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200710 / 37
Early progressexponent4357 37 100 125,000 4,000,000Karl Rubin (UC agstaffBuhler et al.Fermat’s Last Theoremyear 16401753182518391847185719781993PS Breakfast, March 200711 / 37
HeuristicsIf n is large, then a large integer is very unlikely to be an n-thpower.The probability that an bn is an n-th power is less than1/bn 1 .If an bn is an n-th power, then a, b n.So the probability that some an bn is an n-th power, forsome exponent n 4,000,000, is less thanXn 4,000,000Karl Rubin (UC Irvine)X X 1 10 26,000,000 .n 1ba nb aFermat’s Last TheoremPS Breakfast, March 200712 / 37
HeuristicsBy this argument, the chance that Fermat’s Last Theorem isfalse is less than 1 in 26,000,000.This might be enough to convince someone, but it is not a proofof Fermat’s Last Theorem!What if Fermat’s Last Theorem were true just for “probabilistic”reasons, and not for a “structural” reason that could lead to aproof?Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200713 / 37
Elliptic curvesAn elliptic curve is a curve defined by an equation of the formy 2 x 3 Ax 2 Bx Cwith integer constants A, B, C.The elliptic curve y 2 x 3 x was studied by Fermat.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200714 / 37
Elliptic curvesy2 x3 x( 1, 0)Karl Rubin (UC Irvine)(0, 0)Fermat’s Last Theorem(1, 0)PS Breakfast, March 200715 / 37
Elliptic curvesTheorem (Fermat)The only pairs of rational numbers (fractions) x and y thatsatisfy the equationy2 x3 xare (0, 0), (1, 0), and ( 1, 0).Fermat used this fact to prove that a4 b4 c 4 has no wholenumber solutions. It was one of the few complete proofs that hedid fit in the margin of his Diophantus.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200716 / 37
Elliptic curvesProblems mathematicians study about elliptic curves:Given an elliptic curve,–find all solutions in integers x, y ,–find all solutions in rational numbers x, y .Study the collection of all elliptic curves by classifying theirimportant properties.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200717 / 37
Elliptic curves and Fermat’s Last TheoremSuppose Fermat’s Last Theorem is false, so there are a, b, c,and n 3 such that an bn c n . Define an elliptic curveEa,b,c : y 2 x(x an )(x bn ).Idea (Frey, 1985)The elliptic curve Ea,b,c has suchstrange properties that it cannot exist!If correct, Frey’s idea shows that nosuch a, b, c, and n can exist, andhence Fermat’s Last Theorem is true.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200718 / 37
ModularityAn elliptic curve can be modular.Conjecture (Shimura, Taniyama, 1960)Every elliptic curve is modular.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200719 / 37
ModularityTheorem (Ribet, 1986)If an bn c n , then Ea,b,c is notmodular.This finally reduces the truth ofFermat’s Last Theorem to a“structural” question aboutelliptic curves!Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200720 / 37
ModularityTheorem (Wiles , 1994)If A and B are whole numbers,then the elliptic curvey 2 x(x A)(x B)is modular.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200721 / 37
ModularityProof by contradiction:If Fermat’s Last Theorem is false, then there are a, b, c andn 3 such that an bn c n . If so, then:Theorem (Ribet)Theorem (Wiles)Ea,b,c is not modular.Ea,b,c is modular.This contradiction shows that no such a, b, c, n can exist, soFermat’s Last Theorem is true.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200722 / 37
TimelineSummer 1986After Ribet’s work, Wiles begins to work on theShimura-Taniyama conjecture.Spring 1993Wiles completes draft manuscript of his proof.June 21-23, 1993Wiles announces his proof in three lectures on Modularforms, elliptic curves, and Galois representations at aworkshop at the Newton Institue in Cambridge, England.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200723 / 37
The announcementKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200724 / 37
The announcementKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200724 / 37
The announcementKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200724 / 37
The announcementKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200724 / 37
The announcementKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200724 / 37
TimelineSummer 1993A small number of people check Wiles’ manuscript.Autumn 1993Rumors circulate of a “gap” in Wiles’ proof.December 1993Wiles announces gap.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200725 / 37
The “gap”Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200726 / 37
The “gap”Article: 50483 of sci.mathFrom: wiles@rugola.Princeton.EDU (Andrew Wiles)Subject: Fermat statusDate: 4 Dec 93 01:36:50 GMTIn view of the speculation on the status of my work on theTaniyama-Shimura conjecture and Fermat’s Last Theorem I will give abrief account of the situation. During the review process a number ofproblems emerged, most of which have been resolved, but one inparticular I have not yet settled. The key reduction of (most casesof) the Taniyama-Shimura conjecture to the calculation of the Selmergroup is correct. However the final calculation of a precise upperbound for the Selmer group in the semistable case (of the symmetricsquare representation associated to a modular form) is not yetcomplete as it stands. I believe that I will be able to finish thisin the near future using the ideas explained in my Cambridgelectures.The fact that a lot of work remains to be done on themanuscript makes it still unsuitable for release as a preprint.my course in Princeton beginning in February I will give a fullaccount of this work.InAndrew WilesKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200726 / 37
The “gap”Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200726 / 37
TimelineOctober 1994Wiles and Richard Taylorannounce a new joint paper,completing the proof ofFermat’s Last TheoremMay 1995Wiles and Taylor-Wiles papers published in Annals ofMathematicsKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200727 / 37
SuccessKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200728 / 37
SuccessKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200728 / 37
SuccessThe full Shimura-Taniyama conjecture was proved in 1999,using the methods begun by Wiles:Theorem (Breuil, Conrad, Diamond & Taylor, 1999)Every elliptic curve is modular.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200729 / 37
SuccessFermat’s Last Theorem is an important milestone. But muchmore important for the future of mathematics is the substantialprogress Wiles made toward the Shimura-Taniyama Conjecture.The Shimura-Taniyama Conjecture is part of a more generalphilosophy:There are deep and subtle connections between number theoryand other branches of mathematics.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200730 / 37
ModularityA modular form is a function on the unit disk that has specialsymmetries.A cusp form is a modular form that is zero at the “cusps” (certainboundary points).Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200731 / 37
ModularityEvery cusp form gives rise to an elliptic curve ?If an elliptic curve comes from a cusp form in this way, we saythat the elliptic curve is modular.Karl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200732 / 37
Modularity(Ellipticcurves)((Karl Rubin (UC Irvine)“nice” GaloisrepresentationsFermat’s Last TheoremCuspforms))PS Breakfast, March 200733 / 37
Number theory at UCIKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200734 / 37
Elliptic curves are everywhereElliptic curve cryptography is especially well suited for settingswhere space or computing power are limited, such asSmartcardsKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200735 / 37
Elliptic curves are everywhereElliptic curve cryptography is especially well suited for settingswhere space or computing power are limited, such asCell phones and PDA’sKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200735 / 37
Elliptic curves are everywhereElliptic curve cryptography is especially well suited for settingswhere space or computing power are limited, such asDigital postageKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200735 / 37
Elliptic curves are everywhereKarl Rubin (UC Irvine)Fermat’s Last TheoremPS Breakfast, March 200736 / 37
The Solving of Fermat’s Last TheoremKarl RubinEdward and Vivian Thorp Professor of MathematicsMarch 20, 2007Physical Sciences Breakfast LectureKarl Rubin (UC Irvine)Fermat’s Last Theorem1PS Breakfast, March 200737 / 37
Problems mathematicians study about elliptic curves: Given an elliptic curve, –find all solutions in integers x;y, –find all solutions in rational numbers x;y. Study the collection of all elliptic curves by classifying their important properties. Karl Rubin (UC Irvine) Fermat’s Last Theorem PS Breakfast, March 2007 17 / 37
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professor named Andrew Wiles discovered a proof of Fermat’s Last Theorem dealing with algebraic geometry [4]. If all of the work done on Fermat’s Last Theorem by mathematicians through history were compiled into one stack of papers, this stack would be approximately three meters tall [8]. 4. Preliminaries
