An introduction to the theory of numbers

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Foreword By A. Wiles......Page 4
Preface to the Sixth Edition......Page 6
Preface to The Fifth Edition......Page 7
Preface to the First Edition......Page 8
Remarks on Notation......Page 10
Contents......Page 11
11. Divisibility of integers......Page 20
1.2. Prime numbers......Page 21
1.3. Statement of the fundamental theorem of arithmetic......Page 22
1.4. The sequence of primes......Page 23
1.5. Some questions concerning primes......Page 25
1.6. Some notations......Page 26
1.7. The logarithmic function......Page 28
1.8. Statement of the prime number theorem......Page 29
2.2. Further deductions from Euclid's argument......Page 33
2.3. Primes in certain arithmetical progressions......Page 34
2.4. Second proof of Euclid's theorem......Page 36
2.5. Fermat's and Mersenne's numbers......Page 37
2.6. Third proof of Euclid's theorem......Page 39
2.7. Further results on formulae for primes......Page 40
2.9. Moduli of integers......Page 42
2.10. Proof of the fundamental theorem of arithmetic......Page 44
2.11. Another proof of the fundamental theorem......Page 45
3.1. The definition and simplest properties of a Farey series......Page 47
3.2. The equivalence of the two characteristic properties......Page 48
3.3. First proof of Theorems 28 and 29......Page 49
3.4. Second proof of the theorems......Page 50
3.5. The integral lattice......Page 51
3.6: Some simple properties of the fundamental lattice......Page 52
3.7. Third proof of Theorems 28 and 29......Page 54
3.8. The Farey dissection of the continuum......Page 55
3.9. A theorem of Minkowski......Page 56
3.10. Proof of Minkowski's theorem......Page 58
3.11. Developments of Theorem 37......Page 59
4.1. Some generalities......Page 64
4.2. Numbers known to be irrational......Page 65
4.3. The theorem of Pythagoras and its generalizations......Page 66
4.4. The use of the fundamental theorem in the proofs of Theorems 43-45......Page 68
4.5. A historical digression......Page 69
4.6. Geometrical proof of the irrationality of /5......Page 71
4.7. Some more irrational numbers......Page 72
5.1. Highest common divisor and least common multiple......Page 76
5.2. Congruences and classes of residues......Page 77
5.4. Linear congruences......Page 79
5.5. Euler's function ?m)......Page 82
5.6. Applications of Theorems 59 and 61 to trigonometrical sums......Page 84
5.7. A general principle......Page 89
5.8. Construction of the regular polygon of 17 sides......Page 90
6.1. Fermat's theorem......Page 97
6.2. Some properties of binomial coefficients......Page 98
6.3. A second proof of Theorem 72......Page 100
6.4. Proof of Theorem 22......Page 101
6.5. Quadratic residues......Page 102
6.6. Special cases of Theorem 79: Wilson's theorem......Page 104
6.7. Elementary properties of quadratic residues and non-residues......Page 106
6.8. The order of a (mod m)......Page 107
6.9. The converse of Fermat's theorem......Page 108
6.10. Divisibility of 2p- 1 - 1 by p2......Page 110
6.11. Gauss's lemma and the quadratic character of 2......Page 111
6.12. The law of reciprocity......Page 114
6.13. Proof of the law of reciprocity......Page 116
6.14. Tests for primality......Page 117
6.15. Factors of Mersenne numbers; a theorem of Euler......Page 119
7.2. Integral polynomials and identical congruences......Page 122
7.3. Divisibility of polynomials (mod m)......Page 124
7.4. Roots of congruences to a prime modulus......Page 125
7.5. Some applications of the general theorems......Page 127
7.6. Lagrange's proof of Fermat's and Wilson's theorems......Page 129
7.7. The residue of {7 (p - 1) } !......Page 130
7.8. A theorem of Wolstenholme......Page 131
7.9. The theorem of von Staudt......Page 134
7.10. Proof of von Staudt's theorem......Page 135
8.1. Linear congruences......Page 139
8.2. Congruences of higher degree......Page 141
8.3. Congruences to a prime-power modulus......Page 142
8.4. Examples......Page 144
8.5. Bauer's identical congruence......Page 145
8.6. Bauer's congruence: the case p=2......Page 148
8.7. A theorem of Leudesdorf......Page 149
8.8. , Further consequences of Bauer's theorem......Page 151
8.9. The residues of 2P-1 and (p - 1)! to modulus p^2......Page 154
9.1. The decimal associated with a given number......Page 157
9.2. Terminating and recurring decimals......Page 160
9.3. Representation of numbers in other scales......Page 163
9.4. Irrationals defined by decimals......Page 164
9.5. Tests for divisibility......Page 165
9.6. Decimals with the maximum period......Page 166
9.7. Bachet's problem of the weights......Page 168
9.8. The game of Nim......Page 170
9.9. Integers with missing digits......Page 173
9.10. Sets of measure zero......Page 174
9.11. Decimals with missing digits......Page 176
9.12. Normal numbers......Page 177
9.13. Proof that almost all numbers are normal......Page 179
10.1. Finite continued fractions......Page 184
10.2. Convergents to a continued fraction......Page 185
10.3. Continued fractions with positive quotients......Page 187
10.4. Simple continued fractions......Page 188
10.5. The representation of an irreducible rational fraction by a simple continued fraction......Page 189
10.6. The continued fraction algorithm and Euclid's algorithm......Page 191
10.7. The difference between the fraction and its convergents......Page 194
10.8. Infinite simple continued fractions......Page 196
10.9. The representation of an irrational number by an infinite continued fraction......Page 197
10.10. A lemma......Page 199
10.11. Equivalent numbers......Page 200
10.12. Periodic continued fractions......Page 203
10.13. Some special quadratic surds......Page 206
10.14. The series of Fibonacci and Lucas......Page 209
10.15. Approximation by convergents......Page 213
11.1. Statement of the problem......Page 217
11.2. Generalities concerning the problem......Page 218
11.3. An argument of Dirichlet......Page 220
11.4. Orders of approximation......Page 221
11.5. Algebraic and transcendental numbers......Page 222
11.6. The existence of transcendental numbers......Page 224
11.7. Liouville's theorem and the construction of transcendental numbers......Page 225
11.8. The measure of the closest approximations to an arbitrary irrational......Page 227
11.9. Another theorem concerning the convergents to a continued fraction......Page 229
11.10. Continued fractions with bounded quotients......Page 231
11.11. Further theorems concerning approximation......Page 235
11.12. Simultaneous approximation......Page 236
11.13. The transcendence of a......Page 237
11.14. The transcendence of 7r......Page 242
12.1. Algebraic numbers and integers......Page 248
12.2. The rational integers, the Gaussian integers, and the integers of k(p)......Page 249
12.3. Euclid's algorithm......Page 250
12.4. Application of Euclid's algorithm to the fundamental theorem in k(1)......Page 251
12.5. Historical remarks on Euclid's algorithm and the fundamental theorem......Page 253
12.6. Properties of the Gaussian integers......Page 254
12.7. Primes in k(i)......Page 255
12.8. The fundamental theorem of arithmetic in k(i)......Page 257
12.9. The integers of k(p)......Page 260
13.2. The equation x2 + y2 = z2......Page 264
13.3. The equation x4 +y4 = z4......Page 266
13.4. The equation x3 +y3 = z3......Page 267
13.5. The equation x3+y3=3z3......Page 272
13.6. The expression of a rational as a sum of rational cubes......Page 273
13.7. The equation x3+y3+z3=13......Page 276
14.1. Algebraic fields......Page 283
14.2. Algebraic numbers and integers; primitive polynomials......Page 284
14.3. The general quadratic field k(./m)......Page 286
14.4. Unities and primes......Page 287
14.5. The unities of k(J2)......Page 289
14.6. Fields in which the fundamental theorem is false......Page 292
14.7. Complex Euclidean fields......Page 293
14.8. Real Euclidean fields......Page 295
14.9. Real Euclidean fields (continued)......Page 298
15.1. The primes of k(i)......Page 302
15.2. Fermat's theorem in k(i)......Page 304
15.3. The primes of k(p)......Page 305
15.4. The primes of k(J2) and k(,15)......Page 306
15.5. Lucas's test for the primality of the Mersenne number M4n+3......Page 309
15.6. General remarks on the arithmetic of quadratic fields......Page 312
15.7. Ideals in a quadratic field......Page 314
15.8. Other fields......Page 318
16.1. The function 0(n)......Page 321
16.2. A further proof of Theorem 63......Page 322
16.3. The Mobius function......Page 323
16.4. The Mobius inversion formula......Page 324
16.5. Further inversion formulae......Page 326
16.6. Evaluation of Ramanujan's sum......Page 327
16.7. The functions d (n) and ok (n)......Page 329
16.8. Perfect numbers......Page 330
16.9. The function r(n)......Page 332
16.10. Proof of the formula for r(n)......Page 334
17.1. The generation of arithmetical functions by means of Dirichlet series......Page 337
17.2. The zeta function......Page 339
17.3. The behaviour of c(s) when s -+ 1......Page 340
17.4. Multiplication of Dirichlet series......Page 342
17.5. The generating functions of some special arithmetical functions......Page 345
17.6. The analytical interpretation of the Mobius formula......Page 347
17.7. The function A(n)......Page 350
17.8. Further examples of generating functions......Page 353
17.9. The generating function of r(n)......Page 356
17.10. Generating functions of other types......Page 357
18.1. The order of d(n)......Page 361
18.2. The average order of d(n)......Page 366
18.3. The order of a (n)......Page 369
18.4. The order of 0(n)......Page 371
18.5. The average order of 0(n)......Page 372
18.6. The number of squarefree numbers......Page 374
18.7. The order of r(n)......Page 375
19.2. Partitions of numbers......Page 380
19.3. The generating function ofp(n)......Page 381
19.4. Other generating functions......Page 384
19.5. Two theorems of Euler......Page 385
19.6. Further algebraical identities......Page 388
19.7. Another formula for F(x)......Page 390
19.8. A theorem of Jacobi......Page 391
19.9. Special cases of Jacobi's identity......Page 394
19.10. Applications of Theorem 353......Page 397
19.11. Elementary proof of Theorem 358......Page 398
19.12. Congruence properties of p(n)......Page 399
19.13. The Rogers-Ramanujan identities......Page 402
19.14. Proof of Theorems 362 and 363......Page 405
19.15. Ramanujan's continued fraction......Page 408
20.1. Waring's problem: the numbers g(k) and G(k)......Page 412
20.3. Second proof of Theorem 366......Page 414
20.4. Third and fourth proofs of Theorem 366......Page 416
20.5. The four-square theorem......Page 418
20.6. Quaternions......Page 420
20.7. Preliminary theorems about integral quaternions......Page 422
20.8. The highest common right-hand divisor of two quatemions......Page 424
20.9. Prime quaternions and the proof of Theorem 370......Page 426
20.10. The values of g(2) and G(2)......Page 428
20.11. Lemmas for the third proof of Theorem 369......Page 429
20.12. Third proof of Theorem 369: the number of representations......Page 430
20.13. Representations by a larger number of squares......Page 434
21.1. Biquadrates......Page 438
21.2. Cubes: the existence of G(3) and g(3)......Page 439
21.3. A bound forg(3)......Page 441
21.4. Higher powers......Page 443
21.5. A lower bound for g(k)......Page 444
21.6. Lower bounds for G(k)......Page 445
21.7. Sums affected with signs: the number v(k)......Page 450
21.8. Upper bounds for v(k)......Page 452
21.9. The problem of Prouhet and Tarry: the number P(k,j)......Page 454
21.10. Evaluation of P(k,j) for particular k and j......Page 456
21.11. Further problems of Diophantine analysis......Page 459
22.1. The functions t$ (x) and *(x)......Page 470
22.2. Proof that 0 (x) and * (x) are of order x......Page 472
22.3. Bertrand's postulate and a `formula' for primes......Page 474
22.4. Proof of Theorems 7 and 9......Page 477
22.5. Two formal transformations......Page 479
22.6. An important sum......Page 480
22.7. The sum Ep- 1 and the product 1'I (1 - p- 1 )......Page 483
22.8. Mertens's theorem......Page 485
22.9. Proof of Theorems 323 and 328......Page 488
22.10. The number of prime factors of n......Page 490
22.11. The normal order of w(n) and Q (n)......Page 492
22.12. A note on round numbers......Page 495
22.13. The normal order of d(n)......Page 496
22.14. Selberg's theorem......Page 497
22.15. The functions R(x) and V()......Page 500
22.16. Completion of the proof of Theorems 434, 6, and 8......Page 505
22.17. Proof of Theorem 335......Page 508
22.18. Products of k prime factors......Page 509
22.19. Primes in an interval......Page 513
22.20. A conjecture about the distribution of prime pairs p, p + 2......Page 514
23.1. Kronecker's theorem in one dimension......Page 520
23.2. Proofs of the one-dimensional theorem......Page 521
23.3. The problem of the reflected ray......Page 524
23.4. Statement of the general theorem......Page 527
23.5. The two forms of the theorem......Page 529
23.7. Lettenmeyer's proof of the theorem......Page 531
23.8. Estermann's proof of the theorem......Page 533
23.9. Bohr's proof of the theorem......Page 536
23.10. Uniform distribution......Page 539
24.1. Introduction and restatement of the fundamental theorem......Page 542
24.2. Simple applications......Page 543
24.3. Arithmetical proof of Theorem 448......Page 546
24.4. Best possible inequalities......Page 548
24.5. The best possible inequality for i 2 + 172......Page 549
24.6. The best possible inequality for I:; I......Page 551
24.7. A theorem concerning non-homogeneous forms......Page 553
24.8. Arithmetical proof of Theorem 455......Page 555
24.9. Tchebotaref's theorem......Page 556
24.10. A converse of Minkowski's Theorem 446......Page 559
25.1. The congruent number problem......Page 568
25.2. The addition law on an elliptic curve......Page 569
25.3. Other equations that define elliptic curves......Page 575
25.4. Points of finite order......Page 578
25.5. The group of rational points......Page 583
25.6. The group of points modul p.......Page 592
25.7. Integer points on elliptic curves......Page 593
25.8. The L-series of an elliptic curve......Page 597
25.9. Points of finite order and modular curves......Page 601
25.10. Elliptic curves and Fermat's last theorem......Page 605
2. A generalization of Theorem 22......Page 612
3. Unsolved problems concerning primes......Page 613
A LIST OF BOOKS......Page 616
INDEX OF SPECIAL SYMBOLS AND WORDS......Page 620
INDEX OF NAMES......Page 624
GENERAL INDEX......Page 630
Back Cover......Page 642