數(shù)論導(dǎo)引（英文版·第5版）

I. THE SERIES OF PRIMES (1)  
1.1. Divisibility of integers  
1.2. Prime numbers  
1.3. Statement of the fundamental theorem of arithmetic  
1.4. The sequence of primes  
1.5. Some questions concerning primes  
1.6. Some notations  
1.7. The logarithmi function  
1.8. Statement of the prime number theorem  
II. THE SERIES OF PRIMES (2)  
2.1. First proof of Euclid's second theorem  
2.2. Further deductions from Euclid's argument  
2.3. Primes in ertain arithmetical progressions  
2.4. Second proof of Euclid's theorem  
2.5. Fermat's and Mersenne's numbers  
2.6. Third proof of Euclid's theorem  
2.7. Further remarks on formulae for primes  
2.8. Unsolved problems concerning primes  
2.9. Moduli of integers  
2.10. Proof of the fundamental theorem of arithmetic  
2.11. Another proof of the fundamental theorem  
III. FAREY SERIES AND A THEOREM OF MINKOWSKI  
3.1. The definition and simplest properties of a Farey series  
3.2. The equivalence of the two haracteristi properties  
3.3. First proof of Theorems 28 and 29  
3.4. Second proof of the theorems  
3.5. The integral lattice  
3.6. Some simple properties of the fundamental lattice  
3.7. Third proof of Theorems 28 and 29  
3.8. The Farey dissection of the continuum  
3.9. Actheorem of Minkowski  
3.10. Proof of Minkowski's theorem  
3.11. Developments of Theorem 37  
IV. IRRATIONAL NUMBERS  
4.1. Some generalities  
4.2. Numbers known to becirrational  
4.3. The theorem of Pythagoras and its generalizations  
4.4. The use of the fundamental theorem in the proofs of Theorems 43-45  
4.5. A historical digression  
4.6. Geometrical proof of the irrationality of √5  
4.7. Some more irrational numbers  
V. CONGRUENCES AND RESIDUES  
5.1. Highest common divisor and least common multiple  
5.2. Congruences and lasses of residues  
5.3. Elementary properties of congruences  
5.4. Linear congruences  
5.5. Euler's function (m)  
5.6. Applications of Theorems 59 and 61 to trigonometrical sums  
5.7. Acgeneral principle  
5.8. Construction of the regular polygon of 17 sides  
VI. FERMAT'S THEOREM AND ITS CONSEQUENCES  
6.1. Fermat's theorem  
6.2. Some properties of binomial coefficients  
6.3. Acsecond proof of Theorem 72  
6.4. Proof of Theorem 22  
6.5. Quadrati residues  
6.6. Spe ial cases of Theorem 79: Wilson's theorem  
6.7. Elementary properties of quadratic residues and non-residues  
6.8. The order of a (modm)  
6.9. The converse of Fermat's theorem  
6.10. Divisibility of 2p-1 1 by p2  
6.11. Gauss's lemma and the quadratic character of 2  
6.12. The law of reciprocity  
6.13. Proof of the law of reciprocity  
6.14. Tests for primality  
6.15. Factors of Mersenne numbers; a theorem of Euler  
VII. GENERAL PROPERTIES OF CONGRUENCES  
7.1. Roots of ongruences  
7.2. Integral polynomials and identical ongruences  
7.3. Divisibility of polynomials (modm)  
7.4. Roots of congruences to a prime modulus  
7.5. Some applications of the general theorems  
7.6. Lagrange's proof of Fermat's and Wilson's theorems  
7.7. The residue of {1/2(p-1 )} !  
7.8. Actheorem of Wolstenholme  
7.9. The theorem of yon Staudt  
7.10. Proof of yon Staudt's theorem  
VIII. CONGRUENCES TO COMPOSITE MODULI  
8.1. Linear ongruences  
8.2. Congruences of higher degree  
8.3. Congruences to a prime-power modulus  
8.4. Examples  
8.5. Bauer's identical ongruence  
8.6. Bauer's ongruence: the case p=2  
8.7. Actheorem of Leudesdorf  
8.8. Further onsequences of Bauer's theorem  
8.9. The residues of 2p-l and (p-1)! to modulus pZ  
IX. THE REPRESENTATION OF NUMBERS BY DECIMALS  
9.1. The decimal associated with a given number  
9.2. Terminating and recurring decimals  
9.3. Representation of number8 in other scales  
9.4. Irrationals defined by decimals  
9.5. Tests for divisibility  
9.6. Decimals with the maximum period  
9.7. Bachet's problem of the weights  
9.8. The game of Nim  
9.9. Integers with missing digits  
9.10. Sets of measure zero  
9.11. Decimals with missing digits  
9.12. Normal numbers  
9.13. Proof that almost all numbers are normal  
X. CONTINUED FRACTIONS  
10.1. Finite ontinued fractions  
10.2. Convergents to a ontinued fraction  
10.3. Continued fra tions with positive quotients  
10.4. Simple ontinued fractions  
10.5. The representation of an irreducible rational fraction by a simple continued fraction  
10.6. The continued fraction algorithm and Euclid's algorithm  
10.7. The difference between the fraction and its onvergents  
10.8. Infinite simple continued fractions  
10.9. The representation of an irrational number by an infinite continued fraction  
10.10. A lemma  
10.11. Equivalent numbers  
10.12. Periodi continued fractions  
10.13. Some special quadratic surds  
10.14. The series of Fibonacci and Lucas  
10.15. Approximation by convergents  
XI. APPROXIMATION OF IRRATIONALS BY RATIONALS  
11.1. Statement of the problem  
11.2. Generalities on erning the problem  
11.3. An argument of Dirichlet  
11.4. Orders of approximation  
11.5. Algebrai and trans endental numbers  
11.6. The existence of trans endental numbers  
11.7. Liouville's theorem and the construction of transcendental numbers  
11.8. The measure of the closest approximations to an arbitrary irrational  
11.9. Another theorem concerning the convergents to a continued fraction  
11.10. Continued fractions with bounded quotients  
11.11. Further theorems on erning approximation  
11.12. Simultaneous approximation  
11.13. The transcendence of e  
11.14. The transcendence of ∏  
X II. THE FUNDAMENTAL THEOREM OF ARITHMETIC IN k(l), k( i ) , AND k(p )  
12.1. Algebrai numbers and integers  
12.2. The rational integers, the Gaussian integers, and the integers of k(p)  
12.3. Euclid's algorithm  
12.4. Application of Euclid's algorithm to the fundamental theorem  
12.5. Historical remarks on Euclid's algorithm and the fundamental theorem  
12.6. Properties of the Gaussian integers  
12.7. Primes in k(i)  
12.8. The fundamental theorem of arithmeti in k(i)  
12.9. The integers of k(p)  
XIII. SOME DIOPHANTINE EQUATIONS  
13.1. Fermat's last theorem  
13.2. The equation x2+y2=z2  
13.3. The equation x4+y4=z4  
13.4. The equation x3+y3=z3  
13.5. The equation x3+y3=3z3  
13.6. The expression of a rational as a sum of rational ubes  
13.7. The equation x3+y3+z3=t3  
XIV. QUADRATIC FIELDS (1)  
14. I. Algebrai fields  
14.2. Algebrai numbers and integers; primitive polynomials  
14.3. The general quadrati field k(√m)  
14.4. Unities and primes  
14.5. The unities of k(√2)  
14.6. Fields in which the fundamental theoremcis false  
14.7. Complex Euclidean fields    
14.8. Real Euclidean fields  
14.9. Real Euclidean fields (continued)    
XV. QUADRATIC FIELDS (2)  
15.1. The primes of k(i)  
15.2. Fermat's theorem in k(i)  
15.3. The primes of k(p)  
15.4. The primes of k(√2) and k(√5)  
15.5. Lucas's test for the primality of the Mersenne number M4n+s    
15.6. General remarks on the arithmeti of quadrati fields  
15.7. Ideals in a quadrati field    
15.8. Other fields  
XVI. THE ARITHMETICAL FUNCTIONS ~(n), ft(n), d(n), a(n), r(n)  
16.1. The function ∮(n)  
16.2. A further proof of Theorem 63  
16.3. The M6bius function  
16.4. The M6bius inversion formula    
16.5. Further inversion formulae  
16.6. Evaluation of Ramanujan's sum    
16.7. The fun tions d(n) and ak(n)  
16.8. Perfect numbers  
16.9. The fun tion r(n)  
16.10. Proof of the formula for r(n)  
XVII. GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS  
17.1 The generation of arithmeti al fun tions by means of Dirichlet series  
17.2. The zeta function  
17.3. The behaviour of ~(s) when s - 1  
17.4. Multiplication of Dirichlet series    
17.5. The generating functions of some special arithmetical functions  
17.6. The analytical interpretation of the M6bius formula  
17.7. The function A(n)  
17.8. Further examples of generating functions  
17.9. The generating function of r(n)  
17.10. Generating functions of other types  
XVIII. THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS  
18.1. The order of d(n)    
18.2. The average order of d(n)  
18.3. The order of a(n)  
18.4. The order of (n)  
18.5. The average order of ∮(n)  
18.6. The number of squarefree numbers  
18.7. The order of r(n)  
XIX. PARTITIONS  
19.1. The general problem of additive arithmeti  
19.2. Partitions of numbers  
19.3. The generating function of p(n)  
19.4. Other generating functions  
19.5. Two theorems of Euler  
19.6. Further algebrai alcidentities  
19.7. Another formula for F(x)  
19.8. Actheorem of Jacobi  
19.9. Special ases of Jacobi's identity  
19.10. Applications of Theorem 353  
19.11. Elementary proof of Theorem 358  
19.12. Congruen e properties of p(n)  
19.13. The Rogers-Ramanujan identities  
19.14. Proof of Theorems 362 and 363  
19.15. Ramanujan's ontinued fra tion  
XX. THE REPRESENTATION OF A NUMBER BY TWO OR FOUR SQUARES  
20.1. Waring's problem: the numbers g(k) and G(k)  
20.2. Squares  
20.3. Second proof of Theorem 366  
20.4. Third and fourth proofs of Theorem 366  
20.5. The four-square theorem  
20.6. Quaternions  
20.7. Preliminary theorems about integral quaternions  
20.8. The highest common right-hand divisor of two quaternions  
20.9. Prime quaternions and the proof of Theorem 370  
20.10. The values of g(2) and G(2)  
20.11. Lemmas for the third proof of Theorem 369  
20.12. Third proof of Theorem 369: the number of representations  
20.13. Representations by a larger number of squares  
XXI. REPRESENTATION BY CUBES AND HIGHER POWERS  
21.1. Biquadrates  
21.2. Cubes: the existen e of G(3) and g(3)  
21.3. A bound for g(3)  
21.4. Higher powers  
21.5. A lower bound for g(k)  
21.6. Lower bounds for O(k)  
21.7. Sums affe ted with signs: the number v(k)  
21.8. Upper bounds for v(k)  
21.9. The problem of Prouhet and Tarry: the number P(k, j)  
21.10. Evaluation of P(k, j) for parti ular k and j  
21.11. Further problems of Diophantine analysis  
XXII. THE SERIES OF PRIMES (3)  
22.1. The functions tg(x) and ~b(x)  
22.2. Proof that tg(x) and ~b(x) are of order x  
22.3. Bertrand's postulate and a 'formula' for primes  
22.4. Proof of Theorems 7 and 9  
22.5. Two formal transformations  
22.6. An important sum  
22.7. The ∑p-1 and the product ∏ (1--P-1)  
22.8. Mertens's theorem  
22.9. Proof of Theorems 323 and 328  
22.10. The number of prime factors of n  
22.11. The normal order of o(n) and g~(n)  
22.12. A note on round numbers  
22.13. The normal order of d(n)  
22.14. Selberg's theorem  
22.15. The functions R(x) and V(~)  
22.16. Completion of the proof of Theorems 434, 6 and 8  
22.17. Proof of Theorem 335  
22.18. Produ ts of k prime factors  
22.19. Primes in an interval  
22.20. Aconje ture about the distribution of prime pairs p, p+ 2  
XXIII. KRONECKER'S THEOREM  
23.1. Krone ker's theorem in one dimension  
23.2. Proofs of the one-dimensional theorem  
23.3. The problem of the reflected ray  
23.4. Statement of the general theorem  
23.5. The two forms of the theorem  
23.6. An illustration  
23.7. Lettenmeyer's proof of the theorem  
23.8. Estermann's proof of the theorem  
23.9. Bohr's proof of the theorem  
23.10. Uniform distribution    
XXIV. GEOMETRY OF NUMBERS  
24.1. Introduction and restatement of the fundamental theorem  
24.2. Simple applications  
24.3. Arithmetical proof of Theorem  
24.4. Best possible inequalities  
24.5. The best possible inequality for  
24.6. The best possible inequality for  
24.7. Actheorem on erning non-homogeneous forms  
24.8. Arithmetical proof of Theorem  
24.9. Tchebotaref's theorem  
24.10. Aconverse of Minkowski's Theorem  
APPENDIX  
1. Another formula for Pn  
2. Acgeneralization of Theorem 22  
3. Unsolved problems concerning primes  
A LIST OF BOOKS  
INDEX OF SPECIAL SYMBOLS AND WORDS  
INDEX OF NAMES