Ⅰ.THE SERIES OF PRIMES （1） 1
1.1.Divisibility of integers 1
1.2.Prime numbers 1
1.3.Statement of the fundamental theorem of arithmetic 3
1.4.The sequence of primes 3
1.5.Some questions concerning primes 5
1.6.Some notations 7
1.7.The logarithmic function 8
1.8.Statement of the prime number theorem 9
Ⅱ.THE SERIES OF PRIMES （2） 12
2.1.First proof of Euclid's second theorem 12
2.2.Further deductions from Euclid's argument 12
2.3.Primes in certain arithmetical progressions 13
2.4.Second proof of Euclid's theorem 14
2.5.Fermat's and Mersenne's numbers 14
2.6.Third proof of Euclid's theorem 16
2.7.Further remarks on formulae for primes 17
2.8.Unsolved problems concerning primes 19
2.9.Moduli of integers 19
2.10.Proof of the fundamental theorem of arithmetic 21
2.11.Another proof of the fundamental theorem 21
Ⅲ.FAREY SERIES AND A THEOREM OF MINKOWSKI 23
3.1.The definition and simplest properties of a Farey series 23
3.2.The equivalence of the two characteristic properties 24
3.3.First proof of Theorems 28 and 29 24
3.4.Second proof of the theorems 25
3.5.The integral latticc 26
3.6.Some simple properties of the fundamental lattice 27
3.7.Third proof of Theorems 28 and 29 29
3.8.The Farey dissection of the continuum 29
3.9.A theorem of Minkowski 31
3.10.Proof of Minkowski's theorem 32
3.11.Developments of Theorem 37 34
Ⅳ.IRRATIONAL NUMBERS 38
4.1.Some generalities 38
4.2.Numbers known to be irrational 38
4.3.The theorem of Pythagoras and its generalizations 39
4.4.The use of the fundamental theorem in the proofs of Theorems43-45 41
4.5.A historical digression 42
4.6.Geometrical proof of the irrationality of √5 44
4.7.Some more irrational numbers 45
Ⅴ.CONGRUENCES AND RESIDUES 48
5.1.Highest common divisor and least common multiple 48
5.2.Congruences and classes of residues 49
5.3.Elementary properties of congruences 50
5.4.Linear congruences 51
5.5.Euler's function φ（m） 52
5.6.Applications of Theorems 59 and 61 to trigonometrical sums 54
5.7.A general principle 57
5.8.Construction of the regular polygon of 17 sides 57
Ⅵ.FERMAT'S THEOREM AND ITS CONSEQUENCES 63
6.1.Fermat's theorem 63
6.2.Some properties of binomial coefficients 63
6.3.A second proof of Theorem 72 65
6.4.Proof of Theorem 22 66
6.5.Quadratic residues 67
6.6.Special cases of Theorem 79: Wilson's theorem 68
6.7.Elementary properties of quadratic residues and non-residues 69
6.8.The order of a （mod m） 71
6.9.The converse of Fermat's theorem 71
6.10.Divisibility of 2p-1-1 by p2 72
6.11.Gauss's lemma and the quadratic character of 2 73
6.12.The law of reciprocity 76
6.13.Proof of the law of reciprocity 77
6.14.Tests for primality 78
6.15.Factors of Mersenne numbers; a theorem of Euler 80
Ⅶ.GENERAL PROPERTIES OF CONGRUENCES 82
7.1.Roots of congruences 82
7.2.Integral polynomials and identical congruences 82
7.3.Divisibility of polynomials （modm） 83
7.4.Roots of congruences to a prime modulus 84
7.5.Some applications of the general theorems 85
7.6.Lagrange's proof of Fermat's and Wilson's theorems 87
7.7.The residue of {1/2（p- 1）}! 87
7.8.A theorem of Wolstenholme 88
7.9.The theorem of von Staudt 90
7.10.Proof of von Staudt's theorem 91
Ⅷ.CONGRUENCES TO COMPOSITE MODULI 94
8.1.Linear congruences 94
8.2.Congruences of higher degree 95
8.3.Congruences to a prime-power modulus 96
8.4.Examples 97
8.5.Bauer's identical congruence 98
8.6.Bauer's congruence: the case p = 2 100
8.7.A theorem of Leudesdorf 100
8.8.Further consequences of Bauer's theorem 102
8.9.The residues of 2p-1 and （p-1）! to modulus p2 104
Ⅸ.THE REPRESENTATION OF NUMBERS BY DECIMALS 107
9.1.The decimal associated with a given number 107
9.2.Terminating and recurring decimals 109
9.3.Representation of numbers in other scales 111
9.4.Irrationals defined by decimals 112
9.5.Tests for divisibility 114
9.6.Decimals with the maximum period 114
9.7.Bachet's problem of the weights 115
9.8.The game of Nim 117
9.9.Integers with missing digits 120
9.10.Sets of measure zero 121
9.11.Decimals with missing digits 122
9.12.Normal numbers 124
9.13.Proof that almost all numbers are normal 125
Ⅹ.CONTINUED FRACTIONS 129
10.1.Finite continued fractions 129
10.2.Convergents to a continued fraction 130
10.3.Continued fractions with positive quotients 131
10.4.Simple continued fractions 132
10.5.The representation of an irreducible rational fraction by a simple continued fraction 133
10.6.The continued fraction algorithm and Euclid's algorithm 134
10.7.The difference between the fraction and its convergents 136
10.8.Infinite simple continued fractions 138
10.9.The representation of an irrational number by an infinite con-tinued fraction 139
10.10.A lemma 140
10.11.Equivalent numbers 141
10.12.Periodic continued fractions 143
10.13.Some special quadratic surds 146
10.14.The series of Fibonacci and Lucas 148
10.15.Approximation by convergents 151
Ⅺ.APPROXIMATION OF IRRATIONALS BY RATIONALS 154
11.1.Statement of the problem 154
11.2.Generalities concerning the problem 155
11.3.An argument of Dirichlet 156
11.4.Orders of approximation 158
11.5.Algebraic and transcendental numbers 159
11.6.The existence of transcendental numbers 160
11.7.Liouville's theorem and the construction of transcendentalnumbers 161
11.8.The measure of the closest approximations to an arbitrary irrational 163
11.9.Another theorem concerning the convergents to a continued fraction 164
11.10.Continued fractions with bounded quotients 165
11.11.Further theorems concerning approximation 168
11.12.Simultaneous approximation 169
11.13.The transcendence of e 170
11.14.The transcendence of π 173
Ⅻ.THE FUNDAMENTAL THEOREM OF ARITHMETIC IN k（l）,k（i）, AND k（p） 178
12.1.Algebraic numbers and integers 178
12.2.The rational integers, the Gaussian integers, and the integers ofk（p） 178
12.3.Euclid's algorithm 179
12.4.Application of Euclid's algorithm to the fundamental theorem in k（1） 180
12.5.Historical remarks on Euclid's algorithm and the fundamental theorem 181
12.6.Properties of the Gaussian integers 182
12.7.Primes in k（i） 183
12.8.The fundamental theorem of arithmetic in k（i） 185
12.9.The integers of k（p） 187
ⅩⅢ.SOME DIOPHANTINE EQUATIONS 190
13.1.Fermat's last theorem 190
13.2.The equation x2+y2 = z2 190
13.3.The equation x4+y4 = z4 191
13.4.The equation x3+y3 = z3 192
13.5.The equation x3+y3 = 3z3 196
13.6.The expression of a rational as a sum of rational cubes 197
13.7.The equation x3+y3+z3 l3 199
ⅩⅣ.QUADRATIC FIELDS （1） 204
14.1.Algebraic fields 204
14.2.Algebraic numbers and integers; primitive polynomials 205
14.3.The general quadratic field k（√m） 206
14.4.Unities and primes 208
14.5.The unities of k（√2） 209
14.6.Fields in which the fundamental theorem is false 211
14.7.Complex Euclidean fields 212
14.8.Real Euclidean fields 213
14.9.Real Euclidean fields （continued） 215
ⅩⅤ.QUADRATIC FIELDS 218
15.1.The primes of k（i） 218
15.2.Fermat's theorem in k（i） 219
15.3.The primes of k（p） 220
15.4.The primes of k（√2） and k（√5） 221
15.5.Lucas's test for the primality of the Mersenne number M4n+3 223
15.6.General remarks on the arithmetic of quadratic fields 225
15.7.Ideals in a quadratic field 227
15.8.Other fields 230
ⅩⅥ.THE ARITHMETICAL FUNCTIONS φ（n）, μ（n）, d（n）, σ（n）, r（n） 233
16.1.The function φ（n） 233
16.2.A further proof of Theorem 63 234
16.3.The Mobius function 234
16.4.The Mobius inversion formula 236
16.5.Further inversion formulae 237
16.6.Evaluation of Ramanujan's sum 237
16.7.The functions d（n） and σk（n） 239
16.8.Perfect numbers 239
16.9.The function r（n） 241
16.10.Proof of the formula for r（n） 242
ⅩⅦ.GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS 244
17.1 The generation of arithmetical functions by means of Dirichlet series 244
17.2.The zeta function 245
17.3.The behaviour of ξ（s） when s→1 246
17.4.Multiplication of Dirichlet series 248
17.6.The generating functions of some special arithmetical functions 250
17.6.The analytical interpretation of the Mobius formula 251
17.7.The function A（n） 253
17.8.Further examples of generating functions 254
17.9.The generating function of r（n） 256
17.10.Generating functions of other types 257
ⅩⅧ.THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS 260
18.1.The order of d（n） 260
18.2.The average order of d（n） 263
18.3.The order of σ（n） 266
18.4.The order of φ（n） 267
18.5.The average order of φ（n） 268
18.6.The number of squarefree numbers 269
18.7.The order of r（n） 270
ⅩⅨ.PARTITIONS 273
19.1.The general problem of additive arithmetic 273
19.2.Partitions of numbers 273
19.3.The generating function of p（n） 274
19.4.Other generating functions 276
19.5.Two theorems of Euler 277
19.6.Further algebraical identities 280
19.7.Another formula for F（x） 280
19.8.A theorem of Jacobi 282
19.9.Special cases of Jacobi's identity 283
19.10.Applications of Theorem 353 285
19.11.Elementary proof of Theorem 358 286
19.12.Congruence properties of p（n） 287
19.13.The Rogers-Ramanujan identities 290
19.14.Proof of Theorems 362 and 363 292
19.15.Ramanujan's continued fraction 294
ⅩⅩ.THE REPRESENTATION OF A NUMBER BY TWO OR FOUR SQUARES 298
20.1.Waring's problem: the numbers g（k） and G（k） 298
20.2.Squares 299
20.3.Second proof of Theorem 366 299
20.4.Third and fourth proofs of Theorem 366 300
20.5.The four-square theorem 302
20.6.Quaternions 303
20.7.Preliminary theorems about integral quaternions 306
20.8.The highest common right-hand divisor of two quaternions 307
20.9.Prime quaternions and the proof of Theorem 370 309
20.10.The values of g（2） and G（2） 310
20.11.Lemmas for the third proof of Theorem 369 311
20.12.Third proof of Theorem 369: the number of representations 312
20.13.Representations by a larger number of squares 314
ⅩⅪ.REPRESENTATION BY CUBES AND HIGHER POWERS 317
21.1.Biquadrates 317
21.2.Cubes: the existence of G（3） and g（3） 318
21.3.Abound for g（3） 319
21.4.Higher powers 320
21.5.A lower bound for g（k） 321
21.6.Lower bounds for G（k） 322
21.7.Sums affected with signs: the number v（k） 325
21.8.Upper bounds for v（k） 326
21.9.The problem of Prouhet and Tarry: the number P（k, j） 328
21.10.Evaluation of P（k, j） for particular k and j 329
21.11.Further problems of Diophantine analysis 332
ⅩⅫ.THE SERIES OF PRIMES （3） 340
22.1.The functions ？（x） and ？（x） 340
22.2.Proof that ？（x） and ？（x） are of order x 341
22.3.Bertrand's postulate and a ‘formula' for primes 343
22.4.Proof of Theorems 7 and 9 345
22.5.Two formal transformations 346
22.6.An important sum 347
22.7.The sum ∑ p-1 and the product П（1—p-1） 349
22.8.Mertens's theorem 351
22.9.Proof of Theorems 323 and 328 353
22.10.The number of prime factors of n 354
22.11.The normal order of ω（n） and Ω（n） 356
22.12.A note on round numbers 358
22.13.The normal order of d（n） 359
22.14.Selberg's theorem 359
22.15.The functions R（x） and V（ξ） 362
22.16.Completion of the proof of Theorems 434, 6 and 8 365
22.17.Proof of Theorem 335 367
22.18.Products of k prime factors 368
22.19.Primes in an interval 371
22.20.A conjecture about the distribution of prime pairs p, p+2 371
ⅩⅩⅢ.KRONECKER'S THEOREM 375
23.1.Kronecker's theorem in one dimension 375
23.2.Proofs of the one-dimensional theorem 376
23.3.The problem of the reflected ray 378
23.4.Statement of the general theorem 381
23.5.The two forms of the theorem 382
23.6.An illustration 388 384
23.7.Lettenmeyer's proof of the theorem390 387
23.8.Estermann's proof of the theorem 386
23.9.Bohr's proof of the theorem 388
23.10.Uniform distribution 390
ⅩⅩⅣ.GEOMETRY OF NUMBERS 394
24.1.Introduction and restatement of the fundamental theorem 394
24.2.Simple applications 395
24.3.Arithmetical proof of Theorem 448 397
24.4.Best possible inequalities 399
24.5.The best possible inequality for ξ 2+η2 400
24.6.The best possible inequality for |ξη| 401
24.7.A theorem concerning non-homogeneous forms 402
24.8.Arithmetical proof of Theorem 455 405
24.9.Tchebotaref's theorem 405
24.10.A converse of Minkowski's Theorem 446 407
APPENDIX 414
1.Another formula for pn 414
2.A generalization of Theorem 22 414
3.Unsolved problems concerning primes 415
A LIST OF BOOKS 417
INDEX OF SPECIAL SYMBOLS AND WORDS 420
INDEX OF NAMES 423