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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 lattice 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 Theorems 43-45 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 transcendental numbers 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 πr 173

Ⅻ.THE FUNDAMENTAL THEOREM OF ARITHMETIC IN k（1），k（i），AND k（ρ） 178

12.1.Algebraic numbers and integers 178

12.2.The rational integers，the Gaussian integers，and the integers of k（ρ） 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（ρ） 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＝t3 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（2） 218

15.1.The primes of k（i） 218

15.2.Fermat’s theorem in k（i） 219

15.3.The primes of k（ρ） 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

17.5.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 V（x）and ？（x） 340

22.2.Proof that V（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 Π（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 384

23.7.Lettenmeyer’s proof of the theorem 384

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 ξ2＋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