本书是数论领域的一部传世名著，也是现代数学大师哈代的代表作之一。书中作者从多个角度对数论进行了深入阐述，内容包括素数、无理数、同余、费马定理、连分数、不定方程、二次域、算术函数、分划等。新版由第二作者在每章末尾增写了评注，更便于读者阅读。虽然是为数学专业的人士所写，但是大学一年级学生也能读懂。

本书自出版以来一直备受学界推崇，被很多知名大学，如牛津大学、麻省理工学院、加州大学伯克利分校等指定为教材或参考书，也是美国斯坦福大学每个数学与计算机科学专业学生必读的一本书。

本书是一本经典的数论名著的第5版，书的内容成于作者在牛津大学、剑桥大学等大学讲课的讲义，从各个不同角度对数论进行了阐述，包括素数、无理数、同余、Fermat定理、同余式、连分数、不定式、二次域、算术函数、分划等等。第二作者为此书每章增加了必要的注解，便于读者理解并进一步学习。

本书读者对象为大学数学专业学生以及对数论感兴趣的专业人士。

I.　THE SERIES OF PRIMES(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)

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

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

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

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

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

7.1.　Roots of congruences　82

7.2.　Integral polynomials and identical congruences　93

7.3.　Divisibility of polynomials (mod m)　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 yon Staudt　90

7.10.　Proof of yon Staudt's theorem　91

VIII.　CONGRUENCES TO COMPOSITE MODULI

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-l and (p-1)! to modulus pZ　104

IX.　THE REPRESENTATION OF NUMBERS BY DECIMALS

9.1.　The decimal associated with a given number　107

9.2.　Terminating and recurring decimals　109

9.3.　Representation of number8 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

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 continued 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

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 π　173

X II.　THE FUNDAMENTAL THEOREM OF ARITHMETIC IN k(l),k(i),AND k(ρ)

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　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

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)

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)

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+s　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)

16.1.　The function φ(n)　233

16.2.　A further proof of Theorem 63　234

16.3.　The M6bius function　234

16.4.　The M6bius 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

17.1　The generation of arithmetical functions by means of Dirichlet series　244

17.2.　The zeta function　245

17.3. The behaviour of ζ(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 M6bius formula　251

17.7.　The function Λ(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

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

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

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

21.1.　Biquadrates　317

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

21.3.　A bound for g(3)　319

21.4.　Higher powers　320

21.5.　A lower bound for g(k)　321

21.6.　Lower bounds for O(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)

22.1.　The functions υ(x) and Ψ(x)　340

22.2.　Proof thatυ(x) and Ψ(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 ∑p-1 and the product ∏(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

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

24.1.　Introduction and restatement of the fundamental theorem　394

24.2.　Simple applications　395

24.3.　Arithmetical proof of Theorem　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　405

24.9.　Tchebotaref's theorem　405

24.10.　A converse of Minkowski's Theorem　407

APPENDIX

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