Elementary Number Theory in Nine Chapters, Second Edition

444 Pages·2005·2.88 MB·English

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Elementary Number Theory
in Nine Chapters

Elementary Number Theory
in Nine Chapters
Second Edition
JAMES J. TATTERSALL
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Contents
Preface pageix
1 Theintriguingnaturalnumbers
1.1 Polygonalnumbers 1
1.2 Sequencesofnaturalnumbers 23
1.3 Theprincipleofmathematicalinduction 40
1.4 Miscellaneousexercises 43
1.5 Supplementaryexercises 50
2 Divisibility
2.1 Thedivisionalgorithm 55
2.2 Thegreatestcommondivisor 64
2.3 TheEuclideanalgorithm 70
2.4 Pythagoreantriples 76
2.5 Miscellaneousexercises 81
2.6 Supplementaryexercises 84
3 Primenumbers
3.1 Euclidonprimes 87
3.2 Numbertheoreticfunctions 94
3.3 Multiplicativefunctions 103
3.4 Factoring 108
3.5 Thegreatestintegerfunction 112
3.6 Primesrevisited 115
3.7 Miscellaneousexercises 129
3.8 Supplementaryexercises 133
vi
Contents vii
4 Perfectandamicablenumbers
4.1 Perfectnumbers 136
4.2 Fermatnumbers 145
4.3 Amicablenumbers 147
4.4 Perfect-typenumbers 150
4.5 Supplementaryexercises 159
5 Modulararithmetic
5.1 Congruence 161
5.2 Divisibilitycriteria 169
5.3 Euler’sphi-function 173
5.4 Conditionallinearcongruences 181
5.5 Miscellaneousexercises 190
5.6 Supplementaryexercises 193
6 Congruencesofhigherdegree
6.1 Polynomialcongruences 196
6.2 Quadraticcongruences 200
6.3 Primitiveroots 212
6.4 Miscellaneousexercises 222
6.5 Supplementaryexercises 223
7 Cryptology
7.1 Monoalphabeticciphers 226
7.2 Polyalphabeticciphers 235
7.3 Knapsackandblockciphers 245
7.4 Exponentialciphers 250
7.5 Supplementaryexercises 255
8 Representations
8.1 Sumsofsquares 258
8.2 Pell’sequation 274
8.3 Binaryquadraticforms 280
8.4 Finitecontinuedfractions 283
8.5 Infinitecontinuedfractions 291
8.6 p-Adicanalysis 298
8.7 Supplementaryexercises 302
viii Contents
9 Partitions
9.1 Generatingfunctions 304
9.2 Partitions 306
9.3 PentagonalNumberTheorem 311
9.4 Supplementaryexercises 324
Tables
T.1 Listofsymbolsused 326
T.2 Primeslessthan10000 329
T.3 Thevaluesof(cid:1)(n),(cid:2)(n),(cid:3)(n), (cid:4)(n),ø(n),
and(cid:1)(n)fornaturalnumberslessthanor
equalto100 333
Answerstoselectedexercises 336
Bibliography
Mathematics(general) 411
History(general) 412
Chapterreferences 413
Index 421