# 《an introduction to the theory of numbers = 数论导引 英文版 第5版 P435》求取 ⇩

Ⅰ.THE SERIES OF PRIMES （1）1

1.1.Divisibility of integers1

1.2.Prime numbers1

1.3.Statement of the fundamental theorem of arithmetic3

1.4.The sequence of primes3

1.5.Some questions concerning primes5

1.6.Some notations7

1.7.The logarithmic function8

1.8.Statement of the prime number theorem9

Ⅱ.THE SERIES OF PRIMES （2）12

2.1.First proof of Euclid's second theorem12

2.2.Further deductions from Euclid's argument12

2.3.Primes in certain arithmetical progressions13

2.4.Second proof of Euclid's theorem14

2.5.Fermat's and Mersenne's numbers14

2.6.Third proof of Euclid's theorem16

2.7.Further remarks on formulae for primes17

2.8.Unsolved problems concerning primes19

2.9.Moduli of integers19

2.10.Proof of the fundamental theorem of arithmetic21

2.11.Another proof of the fundamental theorem21

Ⅲ.FAREY SERIES AND A THEOREM OF MINKOWSKI23

3.1.The definition and simplest properties of a Farey series23

3.2.The equivalence of the two characteristic properties24

3.3.First proof of Theorems 28 and 2924

3.4.Second proof of the theorems25

3.5.The integral latticc26

3.6.Some simple properties of the fundamental lattice27

3.7.Third proof of Theorems 28 and 2929

3.8.The Farey dissection of the continuum29

3.9.A theorem of Minkowski31

3.10.Proof of Minkowski's theorem32

3.11.Developments of Theorem 3734

Ⅳ.IRRATIONAL NUMBERS38

4.1.Some generalities38

4.2.Numbers known to be irrational38

4.3.The theorem of Pythagoras and its generalizations39

4.4.The use of the fundamental theorem in the proofs of Theorems43-4541

4.5.A historical digression42

4.6.Geometrical proof of the irrationality of √544

4.7.Some more irrational numbers45

Ⅴ.CONGRUENCES AND RESIDUES48

5.1.Highest common divisor and least common multiple48

5.2.Congruences and classes of residues49

5.3.Elementary properties of congruences50

5.4.Linear congruences51

5.5.Euler's function φ（m）52

5.6.Applications of Theorems 59 and 61 to trigonometrical sums54

5.7.A general principle57

5.8.Construction of the regular polygon of 17 sides57

Ⅵ.FERMAT'S THEOREM AND ITS CONSEQUENCES63

6.1.Fermat's theorem63

6.2.Some properties of binomial coefficients63

6.3.A second proof of Theorem 7265

6.4.Proof of Theorem 2266

6.6.Special cases of Theorem 79: Wilson's theorem68

6.7.Elementary properties of quadratic residues and non-residues69

6.8.The order of a （mod m）71

6.9.The converse of Fermat's theorem71

6.10.Divisibility of 2p-1-1 by p272

6.11.Gauss's lemma and the quadratic character of 273

6.12.The law of reciprocity76

6.13.Proof of the law of reciprocity77

6.14.Tests for primality78

6.15.Factors of Mersenne numbers; a theorem of Euler80

Ⅶ.GENERAL PROPERTIES OF CONGRUENCES82

7.1.Roots of congruences82

7.2.Integral polynomials and identical congruences82

7.3.Divisibility of polynomials （modm）83

7.4.Roots of congruences to a prime modulus84

7.5.Some applications of the general theorems85

7.6.Lagrange's proof of Fermat's and Wilson's theorems87

7.7.The residue of {1/2（p- 1）}!87

7.8.A theorem of Wolstenholme88

7.9.The theorem of von Staudt90

7.10.Proof of von Staudt's theorem91

Ⅷ.CONGRUENCES TO COMPOSITE MODULI94

8.1.Linear congruences94

8.2.Congruences of higher degree95

8.3.Congruences to a prime-power modulus96

8.4.Examples97

8.5.Bauer's identical congruence98

8.6.Bauer's congruence: the case p = 2100

8.7.A theorem of Leudesdorf100

8.8.Further consequences of Bauer's theorem102

8.9.The residues of 2p-1 and （p-1）! to modulus p2104

Ⅸ.THE REPRESENTATION OF NUMBERS BY DECIMALS107

9.1.The decimal associated with a given number107

9.2.Terminating and recurring decimals109

9.3.Representation of numbers in other scales111

9.4.Irrationals defined by decimals112

9.5.Tests for divisibility114

9.6.Decimals with the maximum period114

9.7.Bachet's problem of the weights115

9.8.The game of Nim117

9.9.Integers with missing digits120

9.10.Sets of measure zero121

9.11.Decimals with missing digits122

9.12.Normal numbers124

9.13.Proof that almost all numbers are normal125

Ⅹ.CONTINUED FRACTIONS129

10.1.Finite continued fractions129

10.2.Convergents to a continued fraction130

10.3.Continued fractions with positive quotients131

10.4.Simple continued fractions132

10.5.The representation of an irreducible rational fraction by a simple continued fraction133

10.6.The continued fraction algorithm and Euclid's algorithm134

10.7.The difference between the fraction and its convergents136

10.8.Infinite simple continued fractions138

10.9.The representation of an irrational number by an infinite con-tinued fraction139

10.10.A lemma140

10.11.Equivalent numbers141

10.12.Periodic continued fractions143

10.14.The series of Fibonacci and Lucas148

10.15.Approximation by convergents151

Ⅺ.APPROXIMATION OF IRRATIONALS BY RATIONALS154

11.1.Statement of the problem154

11.2.Generalities concerning the problem155

11.3.An argument of Dirichlet156

11.4.Orders of approximation158

11.5.Algebraic and transcendental numbers159

11.6.The existence of transcendental numbers160

11.7.Liouville's theorem and the construction of transcendentalnumbers161

11.8.The measure of the closest approximations to an arbitrary irrational163

11.9.Another theorem concerning the convergents to a continued fraction164

11.10.Continued fractions with bounded quotients165

11.11.Further theorems concerning approximation168

11.12.Simultaneous approximation169

11.13.The transcendence of e170

11.14.The transcendence of π173

Ⅻ.THE FUNDAMENTAL THEOREM OF ARITHMETIC IN k（l）,k（i）, AND k（p）178

12.1.Algebraic numbers and integers178

12.2.The rational integers, the Gaussian integers, and the integers ofk（p）178

12.3.Euclid's algorithm179

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 theorem181

12.6.Properties of the Gaussian integers182

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 EQUATIONS190

13.1.Fermat's last theorem190

13.2.The equation x2+y2 = z2190

13.3.The equation x4+y4 = z4191

13.4.The equation x3+y3 = z3192

13.5.The equation x3+y3 = 3z3196

13.6.The expression of a rational as a sum of rational cubes197

13.7.The equation x3+y3+z3l3199

14.1.Algebraic fields204

14.2.Algebraic numbers and integers; primitive polynomials205

14.4.Unities and primes208

14.5.The unities of k（√2）209

14.6.Fields in which the fundamental theorem is false211

14.7.Complex Euclidean fields212

14.8.Real Euclidean fields213

14.9.Real Euclidean fields （continued）215

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+3223

15.6.General remarks on the arithmetic of quadratic fields225

15.8.Other fields230

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 63234

16.3.The Mobius function234

16.4.The Mobius inversion formula236

16.5.Further inversion formulae237

16.6.Evaluation of Ramanujan's sum237

16.7.The functions d（n） and σk（n）239

16.8.Perfect numbers239

16.9.The function r（n）241

16.10.Proof of the formula for r（n）242

ⅩⅦ.GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS244

17.1The generation of arithmetical functions by means of Dirichlet series244

17.2.The zeta function245

17.3.The behaviour of ξ（s） when s→1246

17.4.Multiplication of Dirichlet series248

17.6.The generating functions of some special arithmetical functions250

17.6.The analytical interpretation of the Mobius formula251

17.7.The function A（n）253

17.8.Further examples of generating functions254

17.9.The generating function of r（n）256

17.10.Generating functions of other types257

ⅩⅧ.THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS260

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 numbers269

18.7.The order of r（n）270

ⅩⅨ.PARTITIONS273

19.1.The general problem of additive arithmetic273

19.2.Partitions of numbers273

19.3.The generating function of p（n）274

19.4.Other generating functions276

19.5.Two theorems of Euler277

19.6.Further algebraical identities280

19.7.Another formula for F（x）280

19.8.A theorem of Jacobi282

19.9.Special cases of Jacobi's identity283

19.10.Applications of Theorem 353285

19.11.Elementary proof of Theorem 358286

19.12.Congruence properties of p（n）287

19.13.The Rogers-Ramanujan identities290

19.14.Proof of Theorems 362 and 363292

19.15.Ramanujan's continued fraction294

ⅩⅩ.THE REPRESENTATION OF A NUMBER BY TWO OR FOUR SQUARES298

20.1.Waring's problem: the numbers g（k） and G（k）298

20.2.Squares299

20.3.Second proof of Theorem 366299

20.4.Third and fourth proofs of Theorem 366300

20.5.The four-square theorem302

20.6.Quaternions303

20.8.The highest common right-hand divisor of two quaternions307

20.9.Prime quaternions and the proof of Theorem 370309

20.10.The values of g（2） and G（2）310

20.11.Lemmas for the third proof of Theorem 369311

20.12.Third proof of Theorem 369: the number of representations312

20.13.Representations by a larger number of squares314

ⅩⅪ.REPRESENTATION BY CUBES AND HIGHER POWERS317

21.2.Cubes: the existence of G（3） and g（3）318

21.3.Abound for g（3）319

21.4.Higher powers320

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 j329

21.11.Further problems of Diophantine analysis332

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 x341

22.3.Bertrand's postulate and a ‘formula' for primes343

22.4.Proof of Theorems 7 and 9345

22.5.Two formal transformations346

22.6.An important sum347

22.7.The sum ∑ p-1 and the product П（1—p-1）349

22.8.Mertens's theorem351

22.9.Proof of Theorems 323 and 328353

22.10.The number of prime factors of n354

22.11.The normal order of ω（n） and Ω（n）356

22.12.A note on round numbers358

22.13.The normal order of d（n）359

22.14.Selberg's theorem359

22.15.The functions R（x） and V（ξ）362

22.16.Completion of the proof of Theorems 434, 6 and 8365

22.17.Proof of Theorem 335367

22.18.Products of k prime factors368

22.19.Primes in an interval371

22.20.A conjecture about the distribution of prime pairs p, p+2371

ⅩⅩⅢ.KRONECKER'S THEOREM375

23.1.Kronecker's theorem in one dimension375

23.2.Proofs of the one-dimensional theorem376

23.3.The problem of the reflected ray378

23.4.Statement of the general theorem381

23.5.The two forms of the theorem382

23.6.An illustration388384

23.7.Lettenmeyer's proof of the theorem390387

23.8.Estermann's proof of the theorem386

23.9.Bohr's proof of the theorem388

23.10.Uniform distribution390

ⅩⅩⅣ.GEOMETRY OF NUMBERS394

24.1.Introduction and restatement of the fundamental theorem394

24.2.Simple applications395

24.3.Arithmetical proof of Theorem 448397

24.4.Best possible inequalities399

24.5.The best possible inequality for ξ 2+η2400

24.6.The best possible inequality for |ξη|401

24.7.A theorem concerning non-homogeneous forms402

24.8.Arithmetical proof of Theorem 455405

24.9.Tchebotaref's theorem405

24.10.A converse of Minkowski's Theorem 446407

APPENDIX414

1.Another formula for pn414

2.A generalization of Theorem 22414

3.Unsolved problems concerning primes415

A LIST OF BOOKS417

INDEX OF SPECIAL SYMBOLS AND WORDS420

INDEX OF NAMES423

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