Springer ribenboim 13 lectures on fermats last theorem (springer 1979)

Paulo Ribenboim 13 Lectures on Fermat's Last Theorem Springer-Verlag New York Heidelberg Berlin Paulo Ribenboim Department of Mathematics and Statistics Jeffery Hall Queen's University Kingston Canada K7L 3N6 AMS Subiect Classifications (1980): 10-03, 12-03, 12Axx Library of Congress Cataloguing in Publication Data Ribenboim, Paulo 13 lectures on Fermat's last theorem Includes bibliographies and indexes Fermat's theorem I Title QA244.R5 512'.74 79-14874 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag @ 1979 by Springer-Verlag New York Inc Printed in the United States of America ISBN 0-387-90432-8 Springer-Verlag New York ISBN 3-540-90432-8 Springer-Verlag Berlin Heidelberg Hommage a AndrC Weil pour sa Leqon: goat, rigueur et pCnCtration Preface Fermat's problem, also called Fermat's last theorem, has attracted the attention of mathematicians for more than three centuries Many clever methods have been devised to attack the problem, and many beautiful theories have been created with the aim of proving the theorem Yet, despite all the attempts, the question remains unanswered The topic is presented in the form of lectures, where I survey the main lines of work on the problem In the first two lectures, there is a very brief description of the early history, as well as a selection of a few of the more representative recent results In the lectures which follow, I examine in succession the main theories connected with the problem The last two lectures are about analogues to Fermat's theorem Some of these lectures were actually given, in a shorter version, at the Institut Henri Poincark, in Paris, as well as at Queen's University, in 1977 I endeavoured to produce a text, readable by mathematicians in general, and not only by specialists in number theory However, due to a limitation in size, I am aware that certain points will appear sketchy Another book on Fermat's theorem, now in preparation, will contain a considerable amount of the technical developments omitted here It will serve those who wish to learn these matters in depth and, I hope, it will clarify and complement the present volume It is for me gratifying to acknowledge the help and encouragement I received in the preparation of this book: A J Coleman and the Mathematics Department at Queen's University-for providing excellent working conditions; E M Wight-for her dilligent and skillful typing of the manuscript; G Cornell-who read the book and helped very much in improving the style; The Canada Council-for partial support; C Pisot and J Oesterlewho arranged for my lectures at the Institut Henri Poincare Vlll Preface It is also my pleasure to report here various suggestions, criticisms and comments from several specialists, whom I consulted on specific points or to whom I have sent an earlier typescript version of this book In alphabetical order: A Baker, D Bertrand, K Inkeri, G Kreisel, H W Lenstra Jr., J M Masley, M Mendes-France, B Mazur, T Metsankyla, A Odlyzko, K Ribet, A Robert, P Samuel, A Schinzel, E Snapper, C L Stewant, G Terjanian, A J van der Poorten, S S Wagstaff, M Waldschmidt, L C Washington Kingston, March, 1979 Paulo Ribenboim General Bibliography There have been several editions of Fermat's works The first printing was supervised by his son Samuel de Fermat 1670 Diophanti Alexandrini Arithmeticorurn libri sex, et de Numeris Multangulis liber unus Cum commentariis C.G Bacheti V.C et observationibus D P de Fermat senatoris Tolosani Accessit Doctrinae Analyticae inventum novum, collectum ex variis ejusdem D de Fermat, epistolis B Bosc, in-folio, Tolosae 1679 Varia Opera Mathematica D Petri de Fermat, Senatoris Tolosani J Pech, in-folio, Tolosae Reprinted in 1861, in Berlin, by Friedlander & Sohn, and in 1969, in Brussels, by Culture et Civilisation 1891/1894/1896/1912/1922 Oeuvres de Fermat, en volumes et un supplement Publikes par les soins de MM Paul Tannery et Charles Henry Gauthier-Villars, Paris J In 1957 the old boys high school of Toulouse was renamed ''Lyck Pierre de Fermat" For the occasion the Toulouse Municipal Library and the Archives of Haute-Garonne organized an exhibit in honor of Fermat A brochure was published, describing considerable "Fermatiana" : 1957 Un Mathematicien de Genie: Pierre de Fermat (1601-1665) Lycee Pierre de Fermat, Toulouse, 1957 Many books, surveys and articles have been devoted totally or in part to a historical or mathematical study of Fermat's work, and more specially, to the last theorem The following selection is based on their interest and availability to the modern reader: 1883 Tannery, P Sur la date des principales decouvertes de Fermat Bull Sci.Math., skr 2,7, 1883, 116-128 Reprinted in Sphinx-Oedipe, 3, 1908, 169-182 X General Bibliography 1860 Smith, H J S Report on the Theory of Numbers, part 11, art 61, Report of the British Association 1860 Collected Mathematical Works, Clarendon Press, Oxford, 1894, 131-13? ~eprintedby Chelsea Publ Co., New York, 1965 1910 Bachmann, P Niedere Zahlentheorie Teubner, Leipzig, 1910 Reprinted by Chelsea Publ CO., New York, 1966 1910 Lind, B Uber das letzte Fermatsche Theorem, Teubner, Leipzig, 1910 1917 Dickson, L E Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers Annals of Math., 18, 1917, 161-187 1919 Bachmann, P Das Fermatproblem in seiner bisherigen Entwicklung, Walter de Gruyter, Berlin, 1919 Reprinted by Springer-Verlag, Berlin, 1976 1920 Dickson, L E History of the Theory of Numbers, 11, Carnegie Institution, Washington, 1920 Reprinted by Chelsea Publ Co., New York, 1971 1921 Mordell, L J Three Lectures on Fermat's Last Theorem, Cambridge University Press, Cambridge, 1921 Reprinted by Chelsea Publ Co., New York, 1962, and by VEB Deutscher Verlag d Wiss Berlin, 1972 1925 Ore, @ Fermats Teorem Norske Mat Tidtskrift 7, 1925, 1-10 1927 Khinchine, A I Velikai Teorema Ferma (The Great Theorem of Fermat) State Editor, MoskowLeningrad, 1927 1928 Vandiver, H S and Wahlin, G E Algebraic Numbers, 11 Bull Nat Research Council, 62, 1928 Reprinted by Chelsea Publ Co., New York, 1967 1934 Monishima, T Fermat's Problem (in Japanese), Iwanami Shoten, Tokyo, 1934, 54 pages 1948 Got, T Une enigme mathematique Le dernier theoreme de Fermat (A chapter in Les Grands Courants de la Penste Mathtmatique, edited by F Le Lionnais) Cahiers du Sud., Marseille, 1948 Reprinted by A Blanchard, Paris, 1962 1961 Bell, E T The Last Problem, Simon and Schuster, New York, 1961 1966 Nogues, R Thkorhe de Fermat, son Histoire, A Blanchard, Paris, 1966 1970 Smadja, R Le Theoreme de Fermat (These de 3' cycle), Universite de Paris VI, 1970 1973 Besenfelder, H J Das Fermat-Problem Diplomarbeit, Universitat Karlsruhe, 61 pages, 1973 1973 Fournier, J C Sur le Dernier Theoreme de Fermat (These de 3ecycle), Universite de Paris VI, 1973 1973 Mahoney, M S The Mathematical Career of Pierre de Fermat, Princeton University Press, Princeton, 1973 General Bibliography 197415 Ferguson, R P On Fermat's Last Theorem, I, 11, 111 J Undergrad Math., 6, 1974, 1-14, 85-98 and 7, 1975, 35-45 1977 Edwards, H M Fermat's Last Theorem A Genetic Introduction to Algebraic Number Theory Springer-Verlag, New York, 1977 For the basic facts about algebraic number theory, the reader may consult: 1966 Borevich, Z I and Shafarevich, I R Number Theory, Academic Press, New York, 1966 1972 Ribenboim, P Algebraic Numbers, Wiley-Interscience, New York 1972 This last book will be quoted as [Ri] ' The sign in front of a bibliography entry indicates that I was unable to examine the item in question All the information gathered in this book stems directly from the original sources 289 Bibliography 1967 Therond, J D L'hypothese de Fermat pour les exposants negatifs Enseignement Math (2), 13, 1967,247-252 1968 Bolker, E D Solutions of Ak + Bk = C* in n x n integral matrices Amer Math Monthly, 75, 1968,759-760 1968 Therond, J D L'hypothese de Fermat pour les exposants negatifs Enseignement Math., (2), 14, 1968, 195-196 1969 Greenleaf, N On Fermat's equation in C (t).Amer Math Monthly, 76, 1969, 808-809 1972 Brenner, J L and de Pillis, J Fermat's equation AP BP= C Pfor matrices of integers Math Mag., 45, 1972, 12-15 1972 Nagell, T Sur la resolubiliti: de Yequation x2 y2 z2 = dans un corps quadratique Acta Arithm., 21, 1972, 35-43 1974 Szymiczek, K Note on a paper by T Nagell Acta Arithm 25, 1974,313-314 + + + Epilogue There is no epilogue The search continues New methods are invented, which will in turn be applied to other problems Or, it is just the reverse And this is the best that could happen, for it is the probing and search of such profound questions that nourishes mathematics Index of Names Abe1,N.H 5,25,29,35,51,53,63,70,226, 238, 239 Adams, J.C 105, 112 Addison, J.W 221 Agoh, T 23, 160 Aigner, A 278,280,282,283,284,285,286, 287, 288 Ankeny, N.C 28,32,33, 132, 133, 136,207, 220 Artin, E 172, 174, 195, 278, 287 Arwin, A 61, 72 Bachet, C.G Bachmann, P 31 Baker, A 26, 232, 233, 234, 235, 236, 237, 238, 240, 243 Bang, A.S 160, 162, 206, 218, 236 Barlow, P 5, 35, 51, 53, 70 Barnett, 1.A 273, 288 Bateman, P.T 56, 74 Beeger,N.G.W.H 23, 152, 163 Bell, E.T 18 Bellman, R 270, 288 Bendz, T.R 69, 71 Bergmann, G 39,49 Bernik, V.I 153 Bernoulli, J 10, 12, 99, 100, 102, 111 Bertrand,L U, 17 Bertrandias, F 183, 197 Birkhoff, G.D 52, 59, 71, 160, 162, 206, 219, 234, 236, 242 Bohnizek, S 206, 219 Bolker, E.D 273, 289 Borevich, Z.I 33, 93 Bourbaki, N 210, 221, 223 Brauer, A 69, 72 Brenner, J.L 273, 289 Brettler, E 251, 261 Brillhart, J 19, 23, 33, 152, 164, 231, 243 Brouncker, W Briickner, H 21, 29, 33, 34, 187, 197, 198 Brumer, A 192, 197 Burnside, W 279, 287 Carlitz, L 63, 74, 105, 106, 112, 132, 136, 186, 197, 204, 220 Carmichael, R.D 24, 31, 41,48, 59, 60,72 Cassels, J.W.S 207,221,222,236,237,242, 243, 28 1, 288 Catalan, E 26, 36, 236, 237, 241 Cauchy, A 6, 13, 14, 17,29,30,35,46,48, 54, 60, 71, 120, 121, 125, 134, 148, 155, 161, 267 debotarev, N.G 180, 195 Chang, C.C 222 Chao KO 236, 243 Chasles, M 13, 17 Chevalley, C 21 Chowla, S 28,32,33,69, 74, 132, 133, 136 Clausen,T 102, 104, 105, 107, 111, 124, 146, 152 Index of Names Cornacchia, G 246, 260 Cramer, G 88 Davenport, H 28, 133 Davis, M 217, 223 Dedekind, R 7, 27, 77, 79, 94, 96, 98 Delange, H 243 Demjanenko, B.A 207, 208, 222 DBnes, P 28, 32, 57, 73, 190, 197, 232, 233, 243 De Pillis, J 273, 289 De Waard, C 18 Dickson, L.E 1, 13,56,62,70, 71, 72, 246, 247, 249, 250, 255, 260, 261 Digby, K 2, 16 Diophantus Dirichlet, G Lejeune , 9,45,48,49,81, 82, 83, 90, 91, 93, 119, 120 Domiaty, R.Z 273, 288 Dubouis, E 31, 48, 162, 261 Duarte, F.J 279, 287 Furtwangler, P 57, 73, 153, 165, 168, 169, 170, 171, 172, 173, 178, 180, 181, 182, 184, 194, 205, 210, 219, 228 Gambioli, D 64, 71 Gandhi, J.M 60, 74, 153, 164, 178, 197 Gauss,C.F , , , 10, 16, 35, 36,39,42, 48,75,86,%, 102, 115, 117, 134, 135, 166, 186, 246, 247 Genocchi, A 6, 29, 30, 121, 125, 134, 148 Germain, Sophie , , 12, 35, 54, 55, 56, 58, 61, 62, 73, 134, 141, 170, 245 Glaisher, J.W.L 105, 112, 155, 162 Godel, K 217 Goldziher, K 69, 72 Gottschalk, E 24, 32, 160, 163 Grassmann, H Grebeniuk, D.G 153, 164 Greenleaf, N 264, 289 Greenberg, R 188 Grey, L.D 66, 73 Griselle, F 66, 73 Gross, F 265, 288 Grosswald, E 48 Edwards, H.M 9, 18 Griin, 107, 112, 188, 196 Eichler,M 21,29,33, 185, 187, 197,206 Griinert, J.A 12, 24, 30, 226, 241 Eisenstein, F.G 8, 12, 126, 153, 167, 170, Gundelfinger, S 171, 172, 173, 193, 194, 198 Gunderson,N.G 19,23,32, 160, 164, 188, Erdos, P 207, 220 202, 220, 230, 231, 242 Emvall, R 204, 222, 223 Gut, M 204, 220 Etherington, I.M.H 273, 287 Euler, L , , , , , , , , , , , Guy, R.K 152, 164 , , , 100, 102, 103, 104, 111, 142, 151, 162,202,203,204,205,208, Hardy, G.H 36, 48, 102, 246, 261 236, 279 Hasse, H 28, 33, 133, 136, 153, 172, 173, Evans T 273, 288 174, 176, 178, 184, 188, 195, 196, 197, 211, 220,221,247,250,251 Hayashi, H 167, 198 Ferentinou-Nicolacopoulou, 59, 60, 74, Heaslet, M.A 107, 112 160, 164 Hecke, E 180, 182, 183, 184, 194 Fermat, Pierre de 1, 2, 3, 16, 35, 36, 37, Hellegouarch, Y 205, 206, 208, 209, 214, 100 215, 221, 222 Fermat, Samuel de Henkin, L 221 Ferrero,B 211,223 Hensel, K 8, 18, 60, 99, 112, 251, 260 Fleck,A 57,72,254 Herbrand, J 188, 196 Flye Sainte-Marie, C 261 Herschel, J.F.W 148, 149, 161 Fogels, E 278, 287 Hilbert, D 14,30,83,91,93, 126, 135,148, Fourier, J.-B 267 162, 171, 180, 181, 194,206,217, 219 Frtnicle de Bessy, B 2, 16 Hofmann, J.E 2, 18 Frobenius, G 1, 23, 31, 57, 72, 139, 153, Holden, H 39, 48 156, 159, 162, 163 Holmboe, B.M 29, 53,70 Fueter,R 126, 135, 170, 171, 194,280,281, Holzer, L 196 282 287 Index of Names Horn, R.A 56, 74 Hua, L.K 260 Hurwitz, A 13, 70, 71, 246, 250, 261 Hyyrii, S 20, 25, 33, 234, 236, 243 Igusa, H 260, 261 Inkeri, K 20,25,26,32,33, 34, 52, 58, 65, 69, 73, 171, 172, 189, 197, 202, 206, 221,227,228,229,234,236,237,238, 239, 242, 243, 254, 261 Itard, J 2, 18 Iwasawa, K 188, 190, 197, 210, 211, 221, 222, 223 Jacobi,C.G.J 11,36, 117, 118, 119, 127, 134, 135, 166, 173,247,248,256,266, 268, 284, 286 James, G 59, 72, 227 Jategaonkar, A.V 265, 288 Jensen,K.L 10,22,31, 106, 107, 110, 112 Jepimaschko 153 Jonquitres, E 64, 71 Kapferer, H 206, 221 Kaplan,P 167, 197 Keisler, H.J 222 Kemeny,I.G 221 Klein, F 16 Klosgen, W 59,74, 153,249,254,255,256, 257, 261 Kobelev, V.V 109, 113, 202, 222 Korkine, A 263, 286 Kotov 13 Krasner, M 29, 32, 57, 73, 115, 141, 148, 149, 150, 159, 163 Kreisel, G 218, 220, 223 Kronecker,L 7, 111, 188, 189 Kubert, D 215, 222, 223 Kubota, T 190, 191, 197 Kummer,E.E , , , , , 1 , , , , 18, 21, 23, 26, 27, 28, 29, 30, 33, 35, 66.71, 75,76, 77, 79, 82, 83, 86, 90, 91,93,98,99, 103, 104, 106, 108, 109, 111,112,113,115, 116, 117, 118,119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131,132,133,134, 135,136, 139, 140, 145, 148, 150, 153, 162, 164, Lagrange, J.L 17, 118 LamO,G , , , 13, 16, 17,35,46,47,48, 71 Landau, E 24, 25, 26, 31, 33, 39, 48, 152, 154, 162, 163, 167, 195, 246, 261 Lang,S 215,221,222,223 Lang, S.-D 134, 137 Langevin, M 236, 243 Lebesgue, V.A 5,46,47,48,236,241,246, 260 Legendre, A.M 4, 5, , 39, 45, 47, 48, 54, 56, 62, 70, 71, 73, 141, 166 Lehmer, D.H 21, 23, 32, 34, 108, 109, 112, 131, 133, 136, 160, 185, 196,201, 202, 220, 230, 231, 232, 242 Lehmer, Emma 21, 23, 32, 63,73, 108, 109, 112, 155, 156, 158, 160,163,201,202, 204, 220, 230, 231, 232, 242 LeLidec, P 147, 148, 164 Leopoldt, H.W 183, 184, 190, 191, 192, 197, 282, 288 Lepisto, T 28, 132, 133, 136 Lerch, M 156, 161, 162 LeVeque, W.J 236, 242, 243, 246, 261 Libri,G 245,246,260 Linkovski, J 153, 164 Liouville, J 6, 7, 9, 13, 17,46,48, 71, 263, 286 Littlewood, J.E 246, 261 Long,L , 3 , 6 , Loxton, J.H 235, 243 Lucas, E 24, 64, 71 Luroth, J 265 Maillet,E 131, 135,205,206,21 Malo, E 135 Manin, Yu I 207, 221, 223 Markoff, V 64, 71 Masley, J.M 28,34,13l, 132, 133, 136, 137 Massoutie, L 69, 72 MatijaseviE, Yu 217, 223 Maury, G 117, 136 Mazur,B 18,208,209,222,223 McDonnell, J 170, 172, 196 Meissner, W 23, 31, 59, 72, 152, 162 Mersenne, M 2, 16, 18, 154, 164 Mertens,F 126, 135 296 Metsankyla, T 22, 33, 34, 107, 113, 132, 136, 186, 198, 204, 211, 222, 223 Mihaljinec, M 69, 73 Mileikowsky, E.N 25, 32, 64, 72 Minc,H 273,288 Minkowski, H 16 Mirimanoff, D 12, 23, 24, 29, 30, 31, 115, 130, 135, 139, 141, 143, 144, 145, 147, 148,151,152,153,154,156,162,163, 164, 165, 169, 171, 177, 178,205, 231 Mitchell, H.H 127, 135 Moller, K 25,32, , , Montel, P 288 Montgomery,H.L 28,33,34, 105, 107, 113, 132, 133, 137 Mordell, L.J 49, 207, 214, 215, 221, 235, 242 Morishima, T 19, 23, 28, 31, 32, 139, 141, 148, 160, 163,185, 188,189, 195,196, 202, 220, 242 Moriya, M 172, 196, 205 Mumford, D 215, 221, 233 Nagell, T 236, 242, 278, 279, 287, 289 Nelson, 154 Newman, M 130, 136 Nickel, L 24, 34, 154, 164 Nicol, C.A 109, 113, 202, 220 Niedermeier, F 66, 73 Nielsen,N 203,219 Noguts, R 18 Noll, C 24, 34, 154, 164 Norlund, N.E 203, 219 Oblith, R 279, 287 Odlyzko, A 206 Oeconomu, G 66, 73 Ogg, A 208, 222 Olbers, W 3, 16 Olson, F.R 132, 136 Orts-Aracil, J.M 271, 272, 288 Pajunen, S 28,34, 132, 137 Pasca1,B 3, 16, 100 Peano, G 216, 217, 273 Pellet, A.E 246, 260, 261 Pipin, T 246, 260 Perez-Cacho, L 20, 33, 58, 69, 73 Perisastri, M 226, 243 Peschl, E 153, 254, 261 Picard, E 266 Index of Names Pisot, C 243 Poincare, H 16 Poitou, G 243 Pollack, B.W 109, 113, 202, 221 Pollaczek, F 23,28,31, 139, 159, 160, 163, 190, 192, 195, 254, 261 Pomey, L 57, 69, 72, 73 Powell, B 206, 223 Puccioni, S 24, 33, 161, 164 Pythagoras Rado, R 102, 112 Rameswar Rao, D 69, 74 Ribenboim,P 81 Ribet, K 189, 198 Riemann, B 10, 13, 94, 190, 203, 21 1, 247 Robinson, A 273, 287 Robinson, Julia 217, 223 Rodrigues-Salinas, B 27 1, 288 Rohrlich, D.E 215, 216, 223 Rosser, B 23, 32, 160, 163, 230, 242 Roth, K.F 25, 232, 233, 236, 243 Rotkiewicz, A 20, 27, 33, 154, 164 Saalschiitz, L 203, 219 Sainte-Croix, Chevalier de 1, 16 Samuel, P 221, 265 Sauer, R 25, 31, 64, 71 Scherk, H.F 203, 218 Schinzel, A 2, 18, 20, 27, 56, 74, 154, 180, 198, 234, 243 Schlichting, F 15, 16 Scholz,A 183, 196,282,287 Schreier, 278, 287 Schruttka von Rechtenstamm, G 131, 136 Schumacher, J 39, 48, 111 Schur, I 246, 261 Schwindt, H 157, 163 Segal, D 59, 73 Selberg, S 236, 242 Selfridge, J.L 109, 113, 202, 220, 221 Serre,J.-P 110,113,208,210,211,221,223 Shafarevich, I.R 33, 93 Shanks, D 49, 110, 113, 264, 288 Shepherdson, J.C 217,221 Siege1,C.L 2 , , , 3 , 106, 110, 113, 132, 136, 232, 233, 234, 236, 242 Sierpinski, W 2, 33, 48, 74, 273, 287 Sims, C 211, 221 Skolem, T 217, 220 Skula, L 29, 33, 187, 188, 197, 198 Slavinsky 154 Index of Names Slavutskii, I.S 184, 197 Smith, H.J.S 17 Sprindiuk, V.G 153 Spunar, V.M 24, 31, 59, 72, 163 Stafford, E 31, 108, 109, 112, 196, 201 Stem, M.A 8, 61, 71 Stevenson, E 260, 261 Stewart, C.L 239, 240, 243 Stickelberger, L 119, 135 Stirling, J 104 Stuff, M 153, 164 Swistak, J.M 69, 74 Sylow, P.L.M 81, 86 Sylvester, J.J 105, 111, 156, 161, 162 Szymicek, K 278 289 Takagi,T 170,171,172,175,176,194,195, 196 Tannery,P 1, 18 Tarski, A 221 Tate, J 222 Terjanian, G 20, 26, 34, 66, 68, 74, 153 Thtrond, J.D 272, 289 Thue, A 232, 233, 236, 242 Tijdeman, R 26, 34, 234, 236, 237, 243 Tonascia, J 19, 23, 33, 152, 164, 231, 243 Tuckermann, B 24, 33, 154, 164 Turan, P 232, 233, 242, 243 Trypanis, A 60, 73 Uchida, K 133, 136 Uspensky, J.V 107, 112 van der Pol, B 270, 287 van der Poorten, A.J 229, 235, 236, 239, 240, 243 Vandiver, H.S l8,2O, 21,23,24,27,28,29, , , , , , , , ,105, 107, 108, 109, 112, 113, 126, 128, 129, 135, 136,139, 150, 154, 155, 156, 157, 160, 162, 163, 164, 170, 172, 178, 183, 184, 185,186,188, 189, 194, 195, 196, 197, 2~,201,202,204,206,219,220,234, 236, 242, 260 Vaughan, R.C 57,74 Vivanti, G 52, 73 von Staudt, C 102, 104, 105, 107, 111, 124, 146, 152 Voronoi,G.F 107, 108 Wagstaff, S.S 19, 21, 22, 34, 60, 74, 109, 110, 113, 199, 202, 211, 223, 238 Wahlin,G 18, 136 Walfisz, A 33 Wallis, J Waring, E 246, 261 Washington,L.C 188, 190, 192, 198, 206, 211, 223 Weber, H 209, 210, 218, 219 Weierstrass, K.T W 207, 213, 214 Weil, A 17, 18, 30, 83, 91, 111, 112 117 Weinberger, P 19,23,33, 152, 164,231,243 Weitkamp, H.M 273, 288 Wieferich, A 12, 23, 24, 27, 31, 63, 139, 151, 152,153, 154, 156, 162, 163, 164, 165, 169, 171, 178, 205, 231 Wendt, E 12, 13, 61, 62, 71 Wilson, J 155, 163, 220 Wolfskehl, P 13, 14, 15, 18 Yamada, K 156, 163, 164 Yamaguchi, 23, 160 Yokoi, H 22, 34, 107, 113 Zassenhaus, H 35 Zinoviev, A 218, 223 Zsigmondy, K 160, 162,206,219,234,236, 24 Subject Index Abel's conjecture 25, 63, 238 relations 53, 54, 226 Baker's theorem 235 Barlow-Abel relations 53 Bernoulli numbers 9, 20, 21, 29, 102 ff generalized 190 polynomials 157 Catalan's equation 26, 236 e c h y ' s theorem Cebotarev 's density theorem 180 Character conductor of a 95 imprimitive 95 modular 95 primitive 95 principal 95 reflected 184 Circulant 61 Class field theory 133 Class number 8, 9, 21, 27, 81, 133 first factor of the 21, 27, 97, 132 real 98 relative 98 second factor of the 20, 27, 97, 133 Cotangent function 104 Cyclotomic equation 15 field 15 integers 76, 78 equation, periods of polynomial 78 integers, prime 77 10, 115 Dedekind's zeta function 94, 96 zeta series 98 Dirichlet's theorem on primes in arithmetic progression 56, 71, 119 Discriminant 78 Divisor 77 Eichler's theorem 21, 29, 185 E-irregularity index 204 Eisenstein's reciprocity law 167 Elliptic curves 207 Entire functions 265 Euler numbers 203 polynomials 203 product 94 Fermat 's congruences 245, 253 Subject Index Fermat's (cont.) curve 212 equation function field 11 last theorem first case 3, 35 second case 4, 35 numbers 2, 3, 154 quotient 22, 63, 152 Frobenius criterion 159 Furtwangler's theorems 169, 170, 182 Galois group 80, 98, 168 r-extensions 21 Gauss quadratic reciprocity law 167 sums 96 Genus 213 Gunderson's estimate 23, 23 Hilbert's tenth problem 217 Hecke's theorem 182, 184 Hensel's lemma 251 Hypothesis H 56 Ideal class 80 Ideal class group 20, 80, 119 p-rank of the 1, 29 Ideal numbers 7, 27, 77 Ideals 7, 27, 78 conjugate 80 equivalent 80 fractional 77, 80 greatest common divisor of 77 least common multiple of 79 norm of 80 maximal 79 prime 79 principal 8, 78 Independent extensions 182 Index 117, 127 Infinite descent 4, 37, 39 Inkeri's estimate 20, 228 theorem 172 Integral basis 78 Irregular couple 109, 110 Irregular field improperly 28, 188 properly 28, 188 Irregularity criterion 105 index 21,22, 29, 109 Iwasawa's theory 210 Jacobi 's cyclotomic function 11, 118, 127, 167 reciprocity law 68 resolvent 118, 119 symbol 67, 167 sums 248 Jensen's theorem 10 Krasner's theorem 149 Kummer's congruences 12, 23, 122, 124, 140, 177 conjecture 132 cyclotomic function 127 extensions 167 lemma on units 8, 86 main (or monumental) theorem 8, 86 ,A-adic logarithm 126 Largest known prime 24, 154 Legendre's symbol 55, 167 theorem Lehmer's (Emma) summation congruences 155, 156, 158 LeLidec polynomials 148 congruences 148 Leopoldt 's conjecture 192 theorem 184, 282 Lerch's summation congruences 156 Local-global principle 251 Logarithmic differential quotients 177 L-series 95 Lucas test 24 Maillet determinant 131 McDonnell's theorem 170 Mersenne numbers 20, 24, 27, 154 Mirimanoff's congruences 145, 178 polynomials 143 theorem 152, 169 Mobius function 69 Subject Index Mordell 's conjecture 14 Mordell - Weil 's theorem 207 Morishima's criterion 160 Moriya's theorems 172 Riemann's hypothesis for function fields 13, 247, 250 zeta function 10, 94, 103, 104 Roth's theorem 233 Nanumbers 273 Nonstandard model Schinzel's conjecture 20, 27, 154 Secant function 202 Semi-primary integer 83, 166 Sentence existential 216 undecidable 217 universal 216 Siege1's theorem 234 Singular integer 171 Sophie Germain's primes 56 theorem 5, 6, 55 Stirling's formula 104 Sums of two squares 36 Symbol Aigner's 284 Jacobi's 67, 167 Legendre's 55, 167 power residue 166 Sylow subgroup 81, 86 17 Obstructing graph 285 integer 283 p-adic L-functions 191 logarithm 192 methods 99 regulator 192 Paris prize 3, 13 Peano's arithmetic 216 Periods 10, 115, 247 Picard's theorem 266 p-independent elements 180 p-integral 103 Pollaczek's criterion 159 Power residue character or symbol Primary integer 181 Prime E-regular 203 irregular 10, 106, 107 ramified 79, 168 regular 8, 21, 86, 90, 106 Sophie Germain 56 split 168 unramified 80, 168 Pythagorean triple 36 Reciprocity law Artin and Hasse's 173 Eisenstein's 167, 173 Gauss' 167 Jacobi's 68, 173 Regularity criterion 99, 103, 108 Regulator 82 Residue power character 127 Residue ring 79 Resolvent , Jacobi's 118, 119 Lagrange's 117, 118 166 Takagi's formula 175, 176 Theory complete 217 consistent 217 Theta functions 268 Totient function 69 Twin primes 134 Units fundamental system of 1, 98 group of 76 Kummer's lemma on 86 Kummer ("Kreiseinheiten") 82, 98, 128 Vandiver's congruence for the first factor of the class number 184 Fermat quotient criterion 159 summation congruence 157 theorem 170 Waring's problem 246 Weierstrass points 213 Subject Index Wendt's determinant 13, 61, 62 theorem 13 Wieferich's theorem 139, 151, 169 Wilson's quotient 155 theorem 155 Wolfskehl prize 13, 14 Zeta function Dedekind's 94, 96 Riemann's 10, 94, 103, 104 Zeta series, Dedekind's 98 ... Variations and Fugue on a Theme Variation I (In the Tone of Polynomial Functions) Variation I1 (In the Tone of Entire Functions) xvi Variation 111 (In the Theta Tone) Variation IV (In the Tone of... Differential Equations) Variation V (Giocoso) Variation VI (In the Negative Tone) Variation VII (In the Ordinal Tone) Variation VIII (In a Nonassociative Tone) Variation IX (In the Matrix Tone) 10 Fugue... Library of Congress Cataloguing in Publication Data Ribenboim, Paulo 13 lectures on Fermat's last theorem Includes bibliographies and indexes Fermat's theorem I Title QA244.R5 512'.74 79-14874