This page intentionally left blank Elementary number theory in nine chapters J A M E S J TAT T E R S A L L Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521585033 © Cambridge University Press 1999 This book is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 1999 ISBN-13 ISBN-10 978-0-511-06583-5 eBook (NetLibrary) 0-511-06583-3 eBook (NetLibrary) ISBN-13 978-0-521-58503-3 hardback ISBN-10 0-521-58503-1 hardback ISBN-13 978-0-521-58531-6 paperback ISBN-10 0-521-58531-7 paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate To Terry Contents Preface vii The intriguing natural numbers 1.1 Polygonal numbers 1.2 Sequences of natural numbers 1.3 The principle of mathematical induction 1.4 Miscellaneous exercises 22 38 41 Divisibility 2.1 The division algorithm 2.2 The greatest common divisor 2.3 The Euclidean algorithm 2.4 Pythagorean triples 2.5 Miscellaneous exercises 49 58 64 70 75 Prime numbers 3.1 Euclid on primes 3.2 Number theoretic functions 3.3 Multiplicative functions 3.4 Factoring 3.5 The greatest integer function 3.6 Primes revisited 3.7 Miscellaneous exercises 79 86 95 100 104 107 122 Perfect and amicable numbers 4.1 Perfect numbers 4.2 Fermat numbers 127 135 iv Contents v 4.3 Amicable numbers 4.4 Perfect-type numbers 137 141 Modular arithmetic 5.1 Congruence 5.2 Divisibility criteria 5.3 Euler's phi-function 5.4 Conditional linear congruences 5.5 Miscellaneous exercises 150 158 162 170 179 Congruences of higher degree 6.1 Polynomial congruences 6.2 Quadratic congruences 6.3 Primitive roots 6.4 Miscellaneous exercises 182 186 198 208 Cryptology 7.1 Monoalphabetic ciphers 7.2 Polyalphabetic ciphers 7.3 Knapsack and block ciphers 7.4 Exponential ciphers 210 219 229 234 Representations 8.1 Sums of squares 8.2 Pell's equation 8.3 Binary quadratic forms 8.4 Finite continued fractions 8.5 In®nite continued fractions 8.6 p-Adic analysis 239 255 261 264 272 279 Partitions 9.1 Generating functions 9.2 Partitions 9.3 Pentagonal Number Theorem 284 286 291 Tables T.1 List of symbols used T.2 Primes less than 10 000 305 308 vi Contents T.3 The values of ô(n), ó(n), ö(n), ì(n), ù(n), and Ù(n) for natural numbers less than or equal to 100 312 Answers to selected exercises 315 Bibliography Mathematics (general) History (general) Chapter references Index 390 391 392 399 Preface Elementary Number Theory in Nine Chapters is primarily intended for a one-semester course for upper-level students of mathematics, in particular, for prospective secondary school teachers The basic concepts illustrated in the text can be readily grasped if the reader has a good background in high school mathematics and an inquiring mind Earlier versions of the text have been used in undergraduate classes at Providence College and at the United States Military Academy at West Point The exercises contain a number of elementary as well as challenging problems It is intended that the book should be read with pencil in hand and an honest attempt made to solve the exercises The exercises are not just there to assure readers that they have mastered the material, but to make them think and grow in mathematical maturity While this is not intended to be a history of number theory text, a genuine attempt is made to give the reader some insight into the origin and evolution of many of the results mentioned in the text A number of historical vignettes are included to humanize the mathematics involved An algorithm devised by Nicholas Saunderson the blind Cambridge mathematician is highlighted The exercises are intended to complement the historical component of the course Using the integers as the primary universe of discourse, the goals of the text are to introduce the student to: the basics of pattern recognition, the rigor of proving theorems, the applications of number theory, the basic results of elementary number theory Students are encouraged to use the material, in particular the exercises, to generate conjectures, research the literature, and derive results either vii viii Preface individually or in small groups In many instances, knowledge of a programming language can be an effective tool enabling readers to see patterns and generate conjectures The basic concepts of elementary number theory are included in the ®rst six chapters: ®nite differences, mathematical induction, the Euclidean Algorithm, factoring, and congruence It is in these chapters that the number theory rendered by the masters such as Euclid, Fermat, Euler, Lagrange, Legendre, and Gauss is presented In the last three chapters we discuss various applications of number theory Some of the results in Chapter and Chapter rely on mathematical machinery developed in the ®rst six chapters Chapter contains an overview of cryptography from the Greeks to exponential ciphers Chapter deals with the problem of representing positive integers as sums of powers, as continued fractions, and p-adically Chapter discusses the theory of partitions, that is, various ways to represent a positive integer as a sum of positive integers A note of acknowledgment is in order to my students for their persistence, inquisitiveness, enthusiasm, and for their genuine interest in the subject The idea for this book originated when they suggested that I organize my class notes into a more structured form To the many excellent teachers I was fortunate to have had in and out of the classroom, in particular, Mary Emma Stine, Irby Cauthen, Esayas Kundert, and David C Kay, I owe a special debt of gratitude I am indebted to Bela Bollobas, Jim McGovern, Mark Rerick, Carol Hartley, Chris Arney and Shawnee McMurran for their encouragement and advice I wish to thank Barbara Meyer, Liam Donohoe, Gary Krahn, Jeff Hoag, Mike Jones, and Peter Jackson who read and made valuable suggestions to earlier versions of the text Thanks to Richard Connelly, Frank Ford, Mary Russell, Richard Lavoie, and Dick Jardine for their help solving numerous computer software and hardware problems that I encountered Thanks to Mike Spiegler, Matthew Carreiro, and Lynn Briganti at Providence College for their assistance Thanks to Roger Astley and the staff at Cambridge University Press for their ®rst class support I owe an enormous debt of gratitude to my wife, Terry, and daughters Virginia and Alexandra, for their in®nite patience, support, and understanding without which this project would never have been completed Bibliography 393 Haggard, P.W and Sadler, B.L, A generalization of triangular numbers, International Journal of Mathematical Education in Science and Technology 25 (1994), 195±202 Hemmerly, H.O., Polyhedral numbers, Mathematics Teacher 66 (1973), 356±62 Hutton, C., Ozanam's Recreational Mathematics and Natural Philosophy, E Riddle, London, 1803 Juzuk, D., Curiosa 56, Scripta Mathematica (1939), 218 Kaprekar, D.R., Another solitaire game, Scripta Mathematica 15 (1949), 244±5 Kay, D.C., On a generalizaton of the Collatz algorithm, Pi Mu Epsilon Journal (1972), 338 Kullman, D.E., Sums of powers of digits, Journal of Recreational Mathematics 14 (1981±82), 4±10 Kumar, V.S., On the digital root series, Journal of Recreational Mathematics 12 (1979±80), 267±70 Larison, C.B., Sidney's series, Fibonacci Quarterly 24 (1986), 313±15 Long, C.T., Gregory interpolation: a trick for problem solvers out of the past, Mathematics Teacher 81 (1983), 323±5 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459±62 Franqui, B and GarcõÂa, M., 57 new multiperfect numbers, Scripta Mathematica 20 (1954), 169±71 GarcõÂa, M., New amicable pairs, Scripta Mathematica 21 (1957), 167±71 Lee, E.J and Madachy, J., The history and discovery of amicable numbers, Journal of Recreational Mathematics (1972), 77±93, 153±73, 231±49 Minoli, D and Bear, R., Hyperperfect numbers, Pi Mu Epsilon Journal (1974), 153±7 Pinch, R.G.E., The Carmichael numbers up to 1015 , Mathematics of Computation 61 (1993), 703±22 Poulet, P., La Chasse aux nombres, pp 9±27, Stevens FreÁres, Bruxelles, 1929 Poulet, P., 43 new couples of amicable numbers, Scripta Mathematica 14 (1948), 77 Prielipp, R.W., Digital sums of perfect numbers and triangular numbers, Mathematics Teacher 62 (1969), 170±82 Prielipp, R.W., Perfect numbers, abundant numbers, and de®cient numbers, Mathematics Teacher 63 (1970) 692±6 Shoemaker, R.W., Perfect Numbers, National Council of Teachers of Mathematics, Reston, VA, 1973 Chapters and Brown, E., The ®rst 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Bibliography 397 Chapter Beutelspacher, A., Cryptology, Mathematical Association of America, Washington, DC, 1994 Dif®e, W and Hellman, M.E., New directions in cryptography, IEEE Transactions on Information Theory IT-22 (1976), 644±54 Herodotus, The History, trans David Grene, University of Chicago Press, Chicago, 1987 Hill, L.S., Concerning certain linear transformations apparatus of cryptography, American Mathematical Monthly 38 (1931), 135±54 Kahn, D., The Codebreakers: the Story of Secret Writing, Macmillan, New York, 1967 Koblitz, N., A Course in Number Theory and Cryptography, Springer-Verlag, New York, 1987 Luciano, D and Prichett, G., Cryptology: from Caesar ciphers to public-key cryptosystems, College Mathematics Journal 18 (1987), 2±17 Merkle, R.C., Secure communications over insecure channels, Communications of the ACM 21 (1978), 294±9 Rivist, R., Shamir, A., and Adleman, L., A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM 21 (1978), 120±6 US Army, Signal Communications Field Manual M-94 Superseded by Field Army Signal Communications FM 11±125, December 1969 Chapter Bachman, G., Introduction to p-adic Numbers and Valuation Theory, Academic Press, New York, 1964 Baker, A., A Concise Introduction to the Theory of Numbers, Cambridge University Press, Cambridge, 1984 Bell, E.T., The Last Problem, Mathematical Association of America, Washington, DC, 1990 Brezinski, C., History of Continued Fractions and Pade Approximations, SpringerVerlag, New York, 1991 Cassels, J.W.S., Rational Quadratic Forms, Academic Press, London, 1978 Conway, J.H., The Sensual Quadratic Form, Mathematical Association of America, Washington, DC, 1997 Edgar, H.M., A First Course in Number Theory, Wadsworth, Belmont, NY, 1988 Edwards, H., Fermat's Last Theorem, Springer-Verlag, New York, 1977 Elkies, N., On A4 B4 C D4 , Mathematics of Computation 51 (1988), 825À35 Ellison, W.J., Waring's problem, American Mathematical Monthly 78 (1981) 10À36 GouveÃa, F.G., p-adic Numbers: an Introduction, Springer, New York, 1991 Lander, L.J and Parkin, T.R., Counterexamples to Euler's conjectures on sums of like powers, Bulletin of the American Mathematical Society 72 (1966), 1079 MacDuf®e, C.C The p-adic numbers of Hensel, American Mathematical Monthly 45 (1938) 500±8; reprinted in Selected Papers in Algebra, Mathematical Association of America, Washington, DC, 1977, 241±9 398 Bibliography Nelson, H.L., A solution to Archimedes' cattle problem, Journal of Recreational Mathematics 13 (1980±81), 164±76 Niven, I., Zuckerman, H.S., and Montgomery, H.L., An Introduction to the Theory of Numbers, 5th ed., Wiley, New York, 1991 Olds, C.D., Continued Fractions, Random House, New York, 1963 Paton, W.R (trans.) The Greek Anthology, vols vol 5, 93±5, Heinemann, London, 1918 Ribenboim P., 13 Lectures on Fermat's Last Theorem, Springer-Verlag, New York, 1979 Ribenboim P., Catalan's Conjecture, Academic Press, Boston, MA., 1994 Robbins, N., Beginning Number Theory, Wm C Brown, Dubuque, IA, 1993 Singh, S., Fermat's Enigma, Walker, New York, 1997 Stanton, R.G., A representation problem, Mathematics Magazine 43 (1970), 130±7 Steiner, R.P., On Mordell's equation y À k x , Mathematics of Computation 46 (1986), 703±14 Strayer, J.K., Elementary Number Theory, PWS, Boston, Mass., 1994 Tausky, O., Sums of squares, American Mathematical Monthly 77 (1970), 805±30 Van der Poorten, A., Notes on Fermat's Last Theorem, Wiley-Interscience, New York, 1996 Vardi, Ilan, Archimedes' Cattle Problem, American Mathematical Monthly 105 (1998), 305±19 Chapter Alder, H.L., Partition identities ± from Euler to the present, American Mathematical Monthly 76 (1969), 733±46 Alder, H.L., The use of generating functions to discuss and prove partition identities, Two-Year College Mathematics Journal 10 (1979), 318±29 Andrews, G.E., The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976 Euler, L Introductio in analysin in®nitorum, M Bousquet, Lausanne, 1748 Introduction to the Analysis of the In®nite, vols., trans John Blanton, SpringerVerlag, New York, 1988, 1990 Hardy, G.H and Ramanujan, S., Asymptotic formulae in combinatorial analysis, Proceedings of the London Mathematical Society (2) 17 (1918), 75±118 Guy, R.K., Two theorems on partitions, Mathematics Gazette 42 (1958), 84±6 Rademacher, H., On the partition function p(n), Proceedings of the London Mathematical Society (2) 43 (1937), 241±57 Ramanujan, S., Proof of certain identities in combinatorial analysis, Proceedings of the Cambridge Philosophical Society 19 (1919), 214±16 Index Note: Classical Chinese names are listed under the ®rst name, which is the surname Names beginning with pre®xes such as de, ibn etc are listed according to custom Abel, Niels Henrik, 17 Abu Bakr Ahmad, 216 Abu Kamil ibn Aslam, 176 abundancy, 147 Adalbold, Bishop of Utrecht, 12 Adleman, Leonard M., 236, 252 Aeneas the Tactician, 211 Aiyer, V Ramaswami, 123 Alaoglu, Leon, 147 Alberti, Leon Battista, 219, 223 Alcuin of York, 128, 176, 177 al-Duraihim, 217 Alexandria, Library of, 13, 79 Alford, W.R., 146 Alfred the Great, 11 algorithms Collatz, 33, 34 Crelle, 202, 203 division, 52 Euclidean, 64 Gauss, 156 Kaprekhar, 34 Moessner, 122 reverse±subtract±reverse±add, 37 aliquot cycle, 91 aliquot part, 91 Allikov, I., 87 al-Qulqashandi, 216, 217 amicable pair, 137ff amicable triple, 140 Amthov, Carl, 256 Andrews, George, 301 Annapurna, U., 91 Apian, Peter, 32 Apollonius of Perga, 255 Apuleius of Madaura, 10 Archimedes, 255±256 Archimedean valuation, 279 Archytas of Tarentum, 127 Argenti, Matteo, 223 Aristagoras, 211 Aristotle, 54 arithmetic function, 86 arithmetic mean, 148 Arianism, 155 Armengaud, Joel, 133, 134 Artin, Emil, 196 conjecture, 204 Aryabhata, 11, 40, 41, 64, 264 Asadulla, Syed, 136 asymptotically equivalent, 109 Augustine, Avicenna, 160 Ayrton, Hertha, 180 Babbage, Charles, 226 Bachet, Claude-Gaspard, 16, 18, 21, 41, 58, 64, 76, 128, 134, 177, 243, 247, 251 Theorem, 170, 171 equation, 260, 261 Bacon, Roger, 218 Baker, Alan, 261 Barbeau, Edward, 47 Barlow, P., 172 Barrow, Isaac, 87 Baudhayana, 256, 257 Bear, Robert, 142 Beaufort, Sir Francis, 226 tableau, 225, 226 Belaso, Giovan Batista, 222, 223 Bellerophon, 211 Bernoulli, Daniel, 18 Bernoulli, Jakob, 41, 46 Bernoulli, Johann, 18, 287 Bernoulli, Nicoloas, 29 399 400 Bertrand, Joseph Louis FrancËois, postulate, 108, 109 Bhaskara, 175, 257, 264, 270, 277, 278 Bhaskara±Brouncker Theorem, 278 Billy, Jacques de, 18 Bianchini, Giovanni, 15 binary-square technique, 154, 155 binary quadratic form, 261 discriminant, 262 positive de®nite, 261 universal, 261 Binet, Jacques-Philippe-Marie, 29 formula, 29±30 Bode, Johann, 23 law, 23, 24, 25 Boethius, 10±12, 128 Boleyn, Ann, 45 Bombelli, Rafael, 15±17, 70, 264 Bourlet, Carlo, 135 Bouton, Charles, 56 Boviles, Charles de (Bouvellus), 83, 129 Brahmagupta, 257 Brancker, Thomas, 101 Brent, R.P., 133 Briggs, Henry, 32 Brooke, Rupert, 284 Brouncker, William (Lord), 18, 257, 277 Brown, Alan, 142 Brozek, Jan (Broscius), 129 Brunn, Viggo, 112 Bruno, Giordano, 223 Caesar Augustus, 212 Caesar, Julius, 212 Caldwell, C., 114, 118 canonical notation, 82 Capella, Martianus, 10 Cardano, Girolamo, 86, 87, 93, 223 Carmichael, R.D 143, 144, 162 lambda function, 179 Carroll, Lewis, see Dodgson, Charles casting out nines, 160 Catalan, E.C., 255 Cataldi, Pietro, 101, 129, 130, 134, 135, 264 Catherine the Great, 183 Cauchy, Augustin-Louis, 17, 114, 167, 172 sequence, 282 central trinomial coef®cient, 22 CesaÁro, Ernesto, 110, 118, 124 Charlemagne, 128 Charles I of Great Britain, 224, 226 Charles II of Great Britain, 224 Charles IX of France, 224 Chaucer, Geoffrey, 11, 218 Chebyshev, P.L., 108, 185 Chen Jing-run, 118 Chernick, J., 146 Index Chester®eld, Lord, 69 Chinese Remainder Theorem, 173ff Chowla, Sarvadaman, 103 function, 92 ciphers af®ne, 214 autokey, 223 block, 231 Caesar, 212 character, 212 digraphic, 231 disk, 220ff exponential, 234ff knapsack, 229ff monoalphabetic, 210ff polyalphabetic, 220 polygraphic, 231 shift, 213 trigraphic, 231 Trithemius, 221, 222 VigeneÁre, 224ff Wheatstone, Charles, 226 wheel, 227 ciphertext, 210 Clark, William, 227 Clarkson, Roland, 133, 134 Clemens, Samuel (Mark Twain), 15 Clement, P., formula, 112 Codes, 210 Cohen, G.L., 133 Cohen, Henri, 91 Cole, Frank Nelson, 132 Collatz, Lothar, 33 Colquitt, Walter, N 133, 134 Colson, John, 87 Columbus, Christopher, 15 common divisor, 58f common multiple, 61f complete residue system, 152 congruence, 150ff Constantine the Great, 155 continued fraction ®nite, 264ff in®nite, 272ff periodic, 275 purely periodic, 275 Conway, John, 23, 46, 263 Cooper, C., 58 Copernicus, 221 coprime, 60 Cotes, Roger, 66, 275 Crelle, A.L., 101, 196, 202 Cromwell, Oliver, 255 crossbones check, 160 Cunningham, Alan, 144, 145 chain, 120 Curie, Marie, 79 401 cyclotomic integer, 252 Cyril, Bishop of Alexandria, 14 Darius I of Persia, 211 Dase, Zacharias, 101 Davenport, Harold, coverings, 155 Davis, Martin, 170 de Bessy, Bernard Frenicle, 71, 89, 92, 93, 94, 95, 103, 130, 143, 165, 259 decryption, 210 Dedekind, Richard, 252 de®ciency, 147 Degenin, C.F., 259 degree, 92 de la ValleÂe-Poussin, C.J., 110, 125, 162 de MeÂreÂ, Chevalier, 30 de Moivre, Abraham, 29 de Morgan, Augustus, 38, 45, 176 Deshoullers, Jean-Marc, 118 Descartes, ReneÂ, 89, 130, 140±143, 148, 246 La GeÂometrie, 87 DeTemple, D., 28 Dickson, L.E., 87, 139 Dif®e, W., 236, 237 Digby, Sir Kenelm, 92 digital root, 34 digital sum, 58 Diophantine equation, 170 Diophantus, 13±16, 43, 45, 72, 170, 239, 241, 246, 248, 251, 253, 256 Arithmetica, 13±18, 22, 239, 251, 253, 256 Dirichlet, P.G Lejeune, 87, 91, 107±9, 115, 168, 246, 251, 252 principle, 179 product, 97 distance function, 280 Dixon, John, 66 Dodgson, Charles (Lewis Carroll), 225, 253 Doyle, Sir Arthur Conan, 239 Donne, John, 182 Dubner, Harry, 114, 118 Duncan, R.L., 91 Durfee, William Pitt, 292, 293 Durfee±Sylvester Theorem, 293 Dyson, F.J., 304 Easter, 155, 156 Easton, Belle, 125 Educational Times, 37, 57, 76, 125, 180 Einstein, Albert, theory of general relativity, 248 Eisenstein, Ferdinand, 193 Elizabeth I of England, 11 Elkies, N.J., 251 elliptical curve, 260 encryption, 210 equivalence relation, 151 Eratosthenes, 79 sieve, 79±80 ErdoÈs, Paul, 110, 147, 155 Ernst, Thomas, 225 Escott, E.B., 115, 139 Essennell, Emma, 57 Euclid, 63, 80±2, 127, 128 Elements, 4, 13, 52, 54, 60, 70, 80±2, 107, 127, 217 algorithm, 64 Lemma, 81 Theorem, 107 Euler, Johannes Albert, 249 Euler, Leonhard, 17, 18, 19, 54, 90, 98, 103, 107, 108, 110, 111, 114, 115, 117, 119, 130, 131, 133, 134, 136, 139, 140, 145, 150, 162, 177, 178, 183, 187, 189, 190, 192, 201, 241, 243, 244, 251, 255, 257, 260, 261, 284, 285, 287, 288, 294, 295, 297, 298 criterion, 189 De fractionibus continuis, 264 Introductio in analysin in®nitorum, 284 parity law, 290 partition formula, 298 pentagonal number theorem, 297 phi-function, 162 product, 111 Euler±Fermat Theorem 165 Euler±Maclaurin Theorem, 87 Euler±Mascheroni constant, 87, 125, 162 extended digital sum, 58 Faggot, Henry, see Bruno Faltings, Gert, 252 Faraday, Michael, 43 Farey, J., 167 fractions, 166, 167 interval, 169 pairs, 167 Faulhaber, Johann, 46 Fechner, Gustav, 29 Felkel, Antonio, 101 Fermat, CleÂment-Samuel, 18 Fermat, Pierre de, 17±19, 22, 71, 92, 93, 102, 103, 130, 131, 135, 136, 138, 140±4, 165, 190, 239, 246, 248, 250, 251, 255, 257, 260, 261 Last Theorem, 18, 114, 188, 251±3 Little Theorem, 165 method of descent, 71 Ferrers, Norman M., 291, 293 diagram, 291ff Theorem, 293 Fibonacci, 3, 26, 39, 40, 129, 158, 175, 178, 239, 241, 264 402 identity, 241 Liber abaci, 15, 26, 101, 129, 158, 175, 241, 264 Liber quadratorum, 3, 4, 29, 39, 239 ®eld, 181 ®nite difference method, Flaccus Albinus, see Alcuin Fortune, Reo, 114 Fouvrey, E., 252 Franklin, Fabian, 295, 296, 299 Theorem, 296 Franqui, Benito, 142 Frenicle, see de Bessy Frey, Gerhard, 252 Friedman, William, 225 Fundamental Theorem of Arithmetic, 82 Gage, Paul, 133, 134 Galbreath, Norman, 115 Gale, Kate, 76 Galileo Galilei, 1, 23, 72, 102, 221 Galois, Evariste, 17, 99, 277 Garcia, Mariano, 139, 142 Gardiner, Vera, 116 Gauss, Karl Friedrich, 17, 25, 65, 82, 108, 109, 114, 136, 137, 150, 156, 162, 164, 165, 173, 182, 184, 187, 191, 193, 201, 202, 204, 206, 246, 247, 251, 252, 261, 264 Disquistiones arithmeticae, 17, 136, 150, 162, 182, 193, 261 Lemma, 191 Quadratic Reciprocity Law general relativity, theory of, 248 Genesis, Book of, 137 GeÂrardin, A., 35, 93, 144 Gerbert of Aurillac, 12, 218 Germain, Sophie, 114, 251, 243 German, R.A., 256 Gergonne, Joseph Diez, 158 Gillies, Don, 132, 134 GIMPS, 133 Girand, Albert, 26 problem, 239 Girand±Euler Theorem, 244 Glaisher, J.W.L., 89, 124 Goldbach, Christian, 114, 117, 243 conjecture, 117 golden ratio, 28±30 golden rectangle, 29 golden triangle, 28 Granville, A., 146 greatest common divisor, 58±59 greatest integer function, 104 Gregorian calendar, 45 Gregory, James, group, 180 Index Gunpowder Plot, 1, 92 Gue Shoujing, Gupta, H., 162 Guy, Richard K., 46, 87, 92, 301 Hadamard, Jacques, 110 Haggard, Paul, 20 Haggis, Peter, 147 Halley, Edmond, 153 comet, 153 Hardy, G.H., 88, 111, 117, 118, 250, 303 harmonic mean, 148 Haro, C.H., 167, 169 Harriot, Thomas, 1, Harrison, John, 32 el-Hasan, 175 Haselgrove, C.B., 92 Hastings, Battle of, 45 Heath-Brown, D.R., 252 Heidel, Wolfgang Ernest, 221 Heilbronn, H.A., 103 Hellman, Martin, 234, 236, 237 Henrietta Maria (Queen of Great Britain), 226 Henry IV of France, 16, 224 Henry VIII of England, 45 Hensel, Kurt, 280 Herodotus, 211 Heron's formula, 77 Hilbert, David, 84, 111, 117, 170, 196, 250 Hildegard von Bingen, 218 Hill, John, 41 Hill, Lester, 231 cipher system, 231 Histiaeus of Miletus, 211 Holtzman, Willhelm, 16, 17 Homer, 211 Horadam, A.F., 78 House of Wisdom, 10, 33 Hrotsvita, 129 Hurwitz, Alexander, 132, 134, 250, 273 Huygens, Christiaan, 264, 269 Hypatia, 14 Hypsicles, 4, 13 Iamblichus of Chalis, 10, 76, 128, 137 Ibn Khaldun, 138, 216 Iliad, 211 index, 206 Index librorum prohibitorum, 221 irrationality of e, 54 of ð, p54 of 2, 54 Isravilov, M.I., 87 Iwaniec, Hendrik, 118 403 Jacobi, Carl Gustav, 89, 109, 206, 244, 248, 299 conjecture, 89 symbol, 196 triple product identity, 299 Jefferson, Thomas, 210, 227 wheel cipher, 227 Jia Xian, 32 Jordan, Barbara, 49 Jumeau, AndreÂ, 141, 142 Juzuk, Dov, 47 Kamasutra, 216 Kanold, H.-J., 133, 147 Kaisiki, F.W., 225 Kaprekar, D.R., 34 algorithm, 34 constant, 34 number, 42 al-Karaji, 15, 32 Kennedy, R.E., 58 Kepler, Johannes, 221 Kersey, John, 87 al-Khalil, 210 Khayyam, Omar, 32 al-Khwaritmi, 33, 158 Korselt, A., 146 Kramp, Christian, Kronecker, L., 89 Kulik, J.P., 101 Kumar, V Sasi, 35 Kummer, Ernst Eduard, 252 Kuratowski, Scott, 133 Ladd Franklin, Christine, 208, 295 Ladies' Diary, 20, 21 Lagrange, Joseph-Louis, 17, 26, 117, 131, 150, 182±184, 187, 188, 248, 262, 275 identity, 16 Theorem, 184, 258 Lambert, Johann, 54, 101, 201, 285 Theorem, 201 LameÂ, Gabriel, 29, 65, 66, 251, 252 formula, 29 Lander, L.J., 251 Landry, F., 131 Larison, Sidney, 26 least common multiple, 61 least residue system, 152 Legendre, Adrien Marie, 17, 104, 109, 114, 115, 150, 187, 188, 192, 246, 247, 251 conjecture, 108 Essai sur la theÂorie des nombres, 188 symbol, 188 Theorem, 105 TheÂorie des nombres, 17, 104 lattice point, 104 Lehman, R.S., 92 Lehmer, D.H., 74, 102, 131, 141, 143±5, 163 Lebesgue, V.A., 172 Leibniz, Gottfried, 16, 43, 145, 165, 184, 287 LentheÂric, P., 75 Leonardo of Pisa, see Fibonacci Lerch, M., 123 Lessing, Gotthold, 255 Levy, Paul, 121 Lewis, Meriweather, 227 Leybourn, Judy, 112 Lindemann, Carl, 264 Liouville, Joseph, 92, 99, 162, 180, 252 formula, 94 lambda-function, 99 Littlewood, J.E., 118, 250 Lucas, Edouard, 20, 27, 32, 41, 130±4, 145, 166 Lucas±Lehmer test, 131±3 Luther, Martin, 13 MacMahon, Percy, 278, 298, 300 Theorem, 300 Magna Carta, 45 Mahler, K., 251 Maillet, E., 250 Malo, E., 145 Marcellus, Marks, Sarah, see Ayrton, Hertha Mazur, Barry, 252 Matiasevich, Yuri, 170 Matsuoka, Y., 26 Maurolico, Francesco (Maurolycus, Franciscus), 129 McDaniel, Wayne, 85 Mead, Margaret, 114 Mengoli, Pietro, 43 Merkle, R.C., 236 Mersenne, Marin, 22, 94, 102, 103, 130, 131, 141±3, 165, 246 primes, 131±4 Mertens, Frantz, conjecture, 98 metric, 280 Metrodorus, 43 Meyer, Margaret, 37 Miller, J.C.P., 206 Minkowski, Hermann, 248 Minoli, Daniel, 142 MoÈbius, A.F., 98 function, 98, 109 inversion formuala, 98, 99 Moessner, Alfred, 122 Mohanty, S.P., 19 Monica set, 85 Montcula, Jean Etienne, 21 Montmort, Pierre ReÂmond de, 32 Moore, E.H., 56 404 Mordell, L.J., 252, 260 conjecture, 252 Morehead, J.C., 136 Moss, T., 173 Motzkin, T.S., 85 mth power residue, 198, 199 MuÈller, Johannes, see Regiomontanus Muqaddimah, 138 NaudeÂ, Philipp, 287 Nelson, Harry, 132, 134 Nemore, Jordanus de (Nemorarius), 32, 129 Neugebauer, Otto, 70 NeuveÂglise, Charles de, 133, 134 Neville, E.H., 88 Newton, Isaac, 8, 17, 87, 214, 274 Nickel, Laura, 132, 134 Niccolo of Brescia, 32 Nicaea, Council of, 155 Nicomachus, 5±7, 10, 19±21, 80, 128, 130, 173 conjectures, 128 Introduction to Arithmetic, 5, 7, 10, 13, 80, 128, 173 Niven, Ivan, 58 Noll, Curt, 132, 134 Novarra, Domenico, 15 number theoretic function, 86 additive, 95 Chowla, 92 completely additive, 95 completely multiplicative, 96 degree, 92 digital root, 34 digital sum, 58 Euler phi-function, 162 extended digital sum, 58 kth powers of the divisors, 90 MoÈbius, 98 multiplicative, 95 number of distinct prime factors, 91 number of divisors, 86 prime counting function, 109 strongly additive, 95 sum of divisors, 86 sum of the aliquot parts, 91 number abundant, 128 almost perfect, 147 amicable, 137 amicable triple, 140 Armstrong, 42 automorphic, 42 Bernoulli, 46, 110 betrothed, 140 Carmichael, 145, 146, 166 composite, 79 Index copperbach, 121 coprime, 60 curious, 42 de®cient, 128 elephantine triple, 76 extraordinary, 42 Euclidean perfect, 127 Fermat, 136±7 Fibonacci, 26 Fibonacci-type, 27 ®gurate, 13 fortunate, 114 h-fold perfect, 148 happy, 25 harmonic, 147 hexagonal, 7, 20 highly composite, 88 Hilbert, 84 isolated prime, 120 k-hyperperfect, 142 k-perfect, 140 k-pseudoprime, 145 k-Smith, 85 k-transposable, 41 Kaprekar, 42 left-truncatable prime, 112±3 Lucas, 27 lucky, 116 m-gonal, m-superperfect, 147 m-triangular, 20 Mersenne prime, 131 minimal, 89 multiperfect, 140 narcissistic, 42 Niven, 58 nth harmonic, 87 nth order ®gurate, 13 oblong, octagonal, 22 octahedral, 21 palindromic, 41 palindromic prime, 113 pandigital, 41 pentagonal, perfect, 127 permutation prime, 113 practical, 146 polite, 77, 124 polygonal, powerful, 42 prime, 79 prime digital sum, 85 prime octet, 112 prime quartet, 112 prime triple, 112 primitive Pythagorean triple, 70, 81 405 primitive semiperfect, 145 pseudoperfect, 145 pseudoprime, 145, 155 pyramidal, 11 Pythagorean, 72 Pythagorean triple, 70 quasiperfect, 147 relatively prime, 60 repunit, 114 reversible prime, 113 right-truncatable prime, 112±3 sad, 25 self, 37 semiperfect, 145 silverbach, 121 Smith, 84 Sophie Germain prime, 114, 252 square, square dance, 46 squarefree, 84 squarefull, 84 star, 19±20 superabundant, 147 superperfect, 147 tetranacci, 36 tetrahedral, 12 triangular, tribonacci, 36 twin prime, 111 Ulam, 116 unitary perfect, 147 unitary amicable, 147±8 unitary nonrepetitive, 146±7 untouchable, 147 weird, 145 zigzag, 46 numeri idonei, 103 Odlyzko, A., 98 Octovian, see Caesar Augustus order of number, 199 Ore, Oystein, 148 Orestes, 14 Ostrowski, A., 280 Oughtred, William, 32 Ozanam, 21 rule, 21 Paganini, Nicolo, 139, 140 Paoli, P., 178 Parady, B., 111 parity, 19 Parkin, T.R., 251 partition of a number, 286±287 of a set, 150, 151 perfect, 300 Pascal, Blaise, 30, 32, 38 Treatise on the Arithmetic Triangle, 30 triangle, 30±2 Paton, W.R., 43 Peirce, C.S., 208 Pell, John, 101, 255 equation, 255ff Peppin, T., 136, 200 test, 136, 200 Pepys, Samuel, 257 Perec, George, 214 Periander of Corinth, 211 Perrin, Emily, 125 Perrone, Oskar, 122 persistence, 35 Pervushin, I., 132, 134 Peurbach, Georges, 15 Philip of Opus, 11 Philip II of Spain, 224 Piazzi, Giuseppe, 25 Pick, G., 124 Pinch, Richard, 146 plaintext, 210 Plato, 5, 54, 74, 94, 127 Academy, 5, 216 Plimpton 322, 70 Plutarch, Polignac, A.A.C.M Prince de, 112, 117 Polya, George, 92 Polybius, 211 Pomerance, Carl, 146, 147 Porges, Arthur, 18, 25 Porta, Giovanni, Battista, 223 Poulet, Paul, 91, 142±4 Powers, R.E., 132, 134 Prime Number Theorem, 109 prime power decomposition, 82 primitive root, 201 principle of mathematical induction, 38 Probus, Valerius, 212 Proetus, King, 211 Ptolemy, 15 Almagest, 14, 15 public-key encryption system, 235ff Putnam, Hillary, 170 Pythagoras of Samos, 1, 4, 70, 74, 137, 140 triangle, 72 triple, 70 Qin Jiushao, 66, 175 Mathematical Treatise in Nine Sections, 173 quadratic nonresidue, 187 quadratic reciprocity law, 193 quadratic residue, 187 quadrivium, 10 al-Qulqashandi, 216, 217 406 Rademacher, Hans, 303 Rahn, Johann Heinrich (Rhonius), 101 Raleigh, Sir Walter, Ramanujan, Srinivasa, 47, 88, 250, 301, 303 sum, 169 RameÂe, Pierre de la (Petrus Ramus), 129 Recorde, Robert, 141±2 reduced residue system, 163 Reeds, James, 221 Regiomontanus (Johannes MuÈller), 15, 129, 178 Regius, Hudalrichus, 129 Regula Nicomachi, relativity, general theory, 248 repunit, 114 residue classes, 150 Ribit, Kenneth, 252 Richert, H.-E., 113 Ridpath, Ian, 153 te Riele, H.J.J., 98, 118, 133 Riemann, Bernhard, 109 hypothesis, 111 zeta-function, 110, 111, 118, 162 Riesel, Hans, 132, 134 ring, 181 Rivist, Ronald, L., 236 Roberval, Gilles Persone de, 22 Robinson, Julia, 170 Robinson, Raphael M., 132, 134 Rogers, Will, 150 Rotkiewicz, A., 145 RSA system, 236±7 rule of the virgins, 177 Sachs, A., 70 Sadler, Bonnie, 20 Saouter, Yannik, 118 Sarrus, F., 145 Saunderson, Nicholas, 66, 87, 274 algorithm, 66±9 Schlegel, V., 177 Schneeberger, William, 263 Schnirelmann, L., 118 Schooten, Fritz van, 87, 101 Schroeder, M.R., 59 Schwenter, Daniel, 267 Seelhof, P., 139 Selberg, Atle, 110 Selfridge, John, 34, 92 Senior Wrangler, 88 sequence Beatty, 349 Chowla, 94 digital root, 35 Fibonacci, 26 Fibonacci-type, 27 Galileo, 23 Index look and say, 23 Lucas, 27 persistence, 35 psi-sequence, 95 Sidney, 26 Sidney product, 26 sociable chain, 91 superincreasing, 25, 229 Serre, Jean-Pierre, 252 Sestri, Giovanni, 226 Shala¯y, A., 163 Shamir, Adi, 236 Shanks, William, 114 shift transformation, 212±13 Shimura, Goro, 252 SierpinÂski, W., 91 sieve of Eratosthenes, 79±80 Simonetta, Cicco, 218 Simson, Robert, 41, 273 Sixtus IV, Pope, 15 skytale, 211 Sloane, Neil, 35 Slowinski, David, 132±4 Sluse, Rene FrancËois de, 32 Smith, H., 85 Smith, J., 111 Smith, Michael, 85 sociable chain, 91 Spencer, Gordon, 133, 134 Speusippus, 11 Srinivasan, A.K., 146 Stanley, Richard, 301 Theorem, 301 Steiner, Ralph, 112 Sterling, James, 88 Stevin, Simon, 16 Stifel, Michael, 13, 32, 74, 87, 129, 158 Strass, E.G., 85 subgroup, 181 subring, 181 Suetonius, 212 Suryanarayana, D., 147 Suzanne set, 85 Synesius of Cyrene, 14 Sylvester II, Pope, see Gerbert Sylvester, J.J., 77, 118, 133, 167, 208, 254, 291, 292, 294 Theorem, 294 Talmud, 158 Tamerlane, 138 Taniyama, Yutaka, 252 Tartaglia, NiccoloÂ, 84, 129 Taylor, Richard, 252 Thabit ibn Qurra, 10, 138, 147 Thales of Miletus, Thanigasalam, V., 251 407 Theodoric, King of the Ostrogoths, 10 Theodorus of Cyrene, 54 Theodosius I, Roman Emperor, 14 Theophilus, Bishop of Alexandria, 14 Theon of Alexandria, 14 Theon of Smyrna, 5, 7, 53, 57, 128, 255 Thrasybulus of Miletus, 211 Thurber, James, 127 Titus, Johann, 23 Torricelli, Evangelista, 102 triangle inequality, 50 Trigg, C.W constant, 37 operator, 37 Tripos, 88 Trithemius, Johannes, 221±2 trivium, 10 Tuckerman, Bryant, 132, 134 Twain, Mark, see Clemens, Samuel Ulam, Stanislaw, 116, 117 numbers, 116 spiral, 124, 125 unitary divisor, 147 Vatsyayana, 216 Vespucci, Amerigo, 15 VieÁte, FrancËois, 16, 70, 224 Vachette, A., 176 valuation Archimedean, 279 non-Archimedean, 279 Vandiver, H.S., 118 Vergil, Aeneid, 18 VigeneÁre, Blaise de, 224 cipher, 224±5 tableau, 224 Vinogradov, I.M., 118, 251 Von Mangolt, H., function, 100 Voroni, Georgi, 171 formula, 171 Wagon, Stan, 163 Walker, Gilbert, 88 Wall, Charles, 147 Wallis, John, 13, 32, 83, 89, 92±4, 224, 257 Arithmetica in®nitorum, 264, 267 Opera mathematica, 255 Walpole, Horace, 69 Walpole, Robert, 69 Wang Xun, ? Wantzel, Pierre Laurent, 129 Waring, Edward, 87, 108, 117, 184, 248, 249 Mediatationes algebraicae, 87, 117, 184, 249 problem, 248, 249, 250 Washington, George, 271 Watkins, William, 136 Watson, G.N., 20 Weierstrass, Karl, 50 Welsh, Luther, 133, 134 Wertheim, G., 21 Western, A.E., 136, 206 Wheatstone, Charles, 226 cipher, 226 Whewell, William, 43, 45 puzzle, 43 Whiston, William, 87 Wilanski, Albert, 84 Wiles, Andrew, 252, 253 Wilkins, John, 224 William and Mary, King and Queen of Great Britain, 224 Williams, H.C., 256 Wilson, John, 184 Theorem, 185 Wolstenholme, J., 186, 198 Woltman, George, 133, 134 Wundt, Wilhelm, 29 Wylie, Alexander, 173 Xylander, see Wilhelm Holtzman Yang Hui, 31 Yi Xing, 175, 178 Young, Alfred, 303 tableau, 303 Zarantonello, S., 111 Zarnke, C.R., 256 Zhang Quijian, 176 Zhu Shijie, ... (general) Chapter references Index 390 391 392 399 Preface Elementary Number Theory in Nine Chapters is primarily intended for a one-semester course for upper-level students of mathematics, in particular,... concepts of elementary number theory are included in the ®rst six chapters: ®nite differences, mathematical induction, the Euclidean Algorithm, factoring, and congruence It is in these chapters... make them think and grow in mathematical maturity While this is not intended to be a history of number theory text, a genuine attempt is made to give the reader some insight into the origin and evolution