An Introduction to the Theory of Numbers, 5th Edition

✍ Scribed by Ivan Morton Niven, Herbert S. Zuckerman, Hugh L. Montgomery

• Publisher: Wiley
• Year: 1991
• Tongue: English
• Leaves: 542
• Edition: 5
• Category: Library

✦ Synopsis

The Fifth Edition of one of the standard works on number theory, written by internationally-recognized mathematicians. Chapters are relatively self-contained for greater flexibility. New features include expanded treatment of the binomial theorem, techniques of numerical calculation and a section on public key cryptography. Contains an outstanding set of problems.

✦ Table of Contents

Cover......Page 1
Title......Page 2
Copyright......Page 3
Preface......Page 4
Contents......Page 6
Notation......Page 10
1.1 INTRODUCTION......Page 14
1.2 DIVISIBILITY......Page 17
PROBLEMS......Page 30
1.3 PRIMES......Page 33
PROBLEMS......Page 41
1.4 THE BINOMIAL THEOREM......Page 48
PROBLEMS......Page 53
NOTES ON CHAPTER 1......Page 57
2.1 CONGRUENCES......Page 60
PROBLEMS......Page 69
2.2 SOLUTIONS OF CONGRUENCES......Page 73
PROBLEMS......Page 75
2.3 THE CHINESE REMAINDER THEOREM......Page 77
PROBLEMS......Page 84
2.4 TECHNIQUES OF NUMERICAL CALCULATION......Page 87
PROBLEMS......Page 95
2.5 PUBLIC-KEY CRYPTOGRAPHY......Page 97
2.6 PRIME POWER MODULI......Page 99
2.7 PRIME MODULUS......Page 104
PROBLEMS......Page 109
2.8 PRIMITIVE ROOTS AND POWER RESIDUES......Page 110
PROBLEMS......Page 119
2.9 CONGRUENCES OF DEGREE TWO, PRIME MODULUS......Page 123
PROBLEMS......Page 127
2.10 NUMBER THEORY FROM AN ALGEBRAIC VIEWPOINT......Page 128
PROBLEMS......Page 132
2.11 GROUPS, RINGS, AND FIELDS......Page 134
PROBLEMS......Page 139
NOTES ON CHAPTER 2......Page 141
3.1 QUADRATIC RESIDUES......Page 144
PROBLEMS......Page 148
3.2 QUADRATIC RECIPROCITY......Page 150
PROBLEMS......Page 153
3.3 THE JACOBI SYMBOL......Page 155
PROBLEMS......Page 160
3.4 BINARY QUADRATIC FORMS......Page 163
PROBLEMS......Page 167
3.5 EQUIVALENCE AND REDUCTION OF BINARY QUADRATIC FORMS......Page 168
PROBLEMS......Page 175
3.6 SUMS OF TWO SQUARES......Page 176
PROBLEMS......Page 182
3.7 POSITIVE DEFINITE BINARY QUADRATIC FORMS......Page 183
NOTES ON CHAPTER 3......Page 189
4.1 GREATEST INTEGER FUNCTION......Page 193
PROBLEMS......Page 197
4.2 ARITHMETIC FUNCTIONS......Page 201
PROBLEMS......Page 204
4.3 THE MOBIUS INVERSION FORMULA......Page 206
PROBLEMS......Page 208
4.4 RECURRENCE FUNCTIONS......Page 210
PROBLEMS......Page 217
4.5 COMBINATORIAL NUMBER THEORY......Page 219
PROBLEMS......Page 223
NOTES ON CHAPTER 4......Page 224
5.1 THE EQUATION ax + by = c......Page 225
PROBLEMS......Page 231
5.2 SIMULTANEOUS LINEAR EQUATIONS......Page 232
PROBLEMS......Page 242
5.3 PYTHAGOREAN TRIANGLES......Page 244
PROBLEMS......Page 246
5.4 ASSORTED EXAMPLES......Page 247
PROBLEMS......Page 252
5.5 TERNARY QUADRATIC FORMS......Page 253
PROBLEMS......Page 261
5.6 RATIONAL POINTS ON CURVES......Page 262
PROBLEMS......Page 273
5.7 ELLIPTIC CURVES......Page 274
PROBLEMS......Page 291
5.8 FACTORIZATION USING ELLIPTIC CURVES......Page 294
PROBLEMS......Page 300
5.9 CURVES OF GENUS GREATER THAN 1......Page 301
NOTES ON CHAPTER 5......Page 302
6.1 FAREY SEQUENCES......Page 310
PROBLEMS......Page 313
6.2 RATIONAL APPROXIMATIONS......Page 314
6.3 IRRATIONAL NUMBERS......Page 320
PROBLEMS......Page 324
6.4 THE GEOMETRY OF NUMBERS......Page 325
PROBLEMS......Page 332
NOTES ON CHAPTER 6......Page 335
7.1 THE EUCLIDEAN ALGORITHM......Page 338
7.2 UNIQUENESS......Page 340
7.3 INFINITE CONTINUED FRACTIONS......Page 342
PROBLEMS......Page 346
7.4 IRRATIONAL NUMBERS......Page 347
7.5 APPROXIMATIONS TO IRRATIONAL NUMBERS......Page 349
PROBLEMS......Page 353
7.6 BEST POSSIBLE APPROXIMATIONS......Page 354
7.7 PERIODIC CONTINUED FRACTIONS......Page 357
7.8 PELL'S EQUATION......Page 364
PROBLEMS......Page 369
7.9 NUMERICAL COMPUTATION......Page 371
NOTES ON CHAPTER 7......Page 372
8.1 ELEMENTARY PRIME NUMBER ESTIMATES......Page 373
PROBLEMS......Page 386
8.2 DIRICHLET SERIES......Page 387
PROBLEMS......Page 400
8.3 ESTIMATES OF ARITHMETIC FUNCTIONS......Page 402
PROBLEMS......Page 411
8.4 PRIMES IN ARITHMETIC PROGRESSIONS......Page 414
PROBLEMS......Page 418
NOTES ON CHAPTER 8......Page 419
CHAPTER 9: Algebraic Numbers......Page 422
9.1 POLYNOMIALS......Page 423
9.2 ALGEBRAIC NUMBERS......Page 427
9.3 ALGEBRAIC NUMBER FI......Page 432
PROBLEMS......Page 436
9.4 ALGEBRAIC INTEGERS......Page 437
9.5 QUADRATIC FIELDS......Page 438
PROBLEMS......Page 440
9.6 UNITS IN QUADRATIC FIELDS......Page 441
9.7 PRIMES IN QUADRATIC FIELDS......Page 442
9.8 UNIQUE FACTORIZATION......Page 444
PROBLEMS......Page 445
9.9 PRIMES IN QUADRATIC FIELDS HAVING THE UNIQUE FACTORIZATION PROPERTY......Page 446
PROBLEMS......Page 453
9.10 THE EQUATION x 3 + y3 = Z3......Page 454
NOTES ON CHAPTER 9......Page 458
10.1 PARTITIONS......Page 459
PROBLEMS......Page 460
10.2 FERRERS GRAPHS......Page 461
PROBLEMS......Page 463
10.3 FORMAL POWER SERIES, GENERATING FUNCTIONS, AND EULER'S IDENTITY......Page 465
10.4 EULER'S FORMULA; BOUNDS ON p(n)......Page 470
10.5 JACOBI'S FORMULA......Page 476
NOTES ON CHAPTER 10......Page 484
CHAPTER 11: The Density of Sequences of Integers......Page 485
11.1 ASYMPTOTIC DENSITY......Page 486
PROBLEMS......Page 488
11.2 SCHNIRELMANN DENSITY AND THE aß THEOREM......Page 489
NOTES ON CHAPTER 11......Page 494
A.I THE FUNDAMENTAL THEOREM OF ALGEBRA......Page 495
PROBLEMS......Page 502
A.3 A SPECIAL VALUE OF THE RIEMANN ZETA FUNCTION......Page 503
PROBLEMS......Page 504
A.4 LINEAR RECURRENCES......Page 506
PROBLEMS......Page 511
General References......Page 513
Hints......Page 516
Answers......Page 525
Index......Page 535