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
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