23, 1993, Andrew Wiles wrote on a blackboard, before an audience at the Newton Institute in Cambridge, England, that if p is a prime number, u, v, and w are rational numbers, and up + vp + wp = 0, then uvw = 0. In other words, he announced that he could prove Fermatâ€™s Last Theorem. His announce- ment came at the end of his series of three talks entitled â€œModular forms, elliptic curves, and Galois representationsâ€� at the week-long workshop on â€œp-adic Galois representations, Iwasawa theory, and the Tamagawa numbers of motivesâ€�.\n\n[\n\n1 v 0 2 2 7 0 4 9 / h t a m : v i X r a\n\nIn the margin of his copy of the works of Diophantus, next to a problem on\n\nPythagorean triples, Pierre de Fermat (1601â€“1665) wrote:\n\nCubum autem in duos cubos, aut quadratoquadratum in duos quadrato- quadratos, et generaliter nullam in inï¬�nitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere : cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.\n\n(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)\n\nWe restate Fermatâ€™s conjecture as follows.\n\nFermatâ€™s Last Theorem. If n > 2, then an +bn = cn has no solutions in nonzero integers a, b, and c.\n\nA proof by Fermat has never been found, and the problem has remained open, inspiring many generations of mathematicians. Much of modern number theory has been built on attempts to prove Fermatâ€™s Last Theorem. For details on the\n\nReceived by the editors November 29, 1993. 1991 Mathematics Subject Classiï¬�cation. Primary 11G05; Secondary 11D41, 11G18. The authors thank the National Science Foundation for ï¬�nancial support.\n\nc(cid:13)1994 American Mathematical Society 0273-0979/94 $1.00 + $.25 per page\n\n1\n\n2\n\nK. RUBIN AND A. SILVERBERG\n\nhistory of
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: 23, 1993, Andrew Wiles wrote on a blackboard, before an audience at the Newton Institute in Cambridge, England, that if p is a prime number, u, v, and w are rational numbers, and up + vp + wp = 0, then uvw = 0. In other words, he announced that he could prove Fermatâ€™s Last Theorem. His announce- ment came at the end of his series of three talks entitled â€œModular forms, elliptic curves, and Galois representationsâ€� at the week-long workshop on â€œp-adic Galois representations, Iwasawa theory, and the Tamagawa numbers of motivesâ€�.\n\n[\n\n1 v 0 2 2 7 0 4 9 / h t a m : v i X r a\n\nIn the margin of his copy of the works of Diophantus, next to a problem on\n\nPythagorean triples, Pierre de Fermat (1601â€“1665) wrote:\n\nCubum autem in duos cubos, aut quadratoquadratum in duos quadrato- quadratos, et generaliter nullam in inï¬�nitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere : cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.\n\n(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)\n\nWe restate Fermatâ€™s conjecture as follows.\n\nFermatâ€™s Last Theorem. If n > 2, then an +bn = cn has no solutions in nonzero integers a, b, and c.\n\nA proof by Fermat has never been found, and the problem has remained open, inspiring many generations of mathematicians. Much of modern number theory has been built on attempts to prove Fermatâ€™s Last Theorem. For details on the\n\nReceived by the editors November 29, 1993. 1991 Mathematics Subject Classiï¬�cation. Primary 11G05; Secondary 11D41, 11G18. The authors thank the National Science Foundation for ï¬�nancial support.\n\nc(cid:13)1994 American Mathematical Society 0273-0979/94 $1.00 + $.25 per page\n\n1\n\n2\n\nK. RUBIN AND A. SILVERBERG\n\nhistory of