Book:G.H. Hardy/An Introduction to the Theory of Numbers/Fifth Edition
Jump to navigation
Jump to search
G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th Edition)
Published $\text {1979}$, Oxford University Press
- ISBN 0-19-853171-0
Subject Matter
Contents
- Preface to the Fifth Edition (Aberdeen, October 1978, E.M.W)
- REMARKS ON NOTATION
- $\text {I}$. 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 logarithmic function
- 1.8. Statement of the prime number theorem
- $\text {II}$. 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 certain 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
- $\text {III}$. 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 characteristic 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. A theorem of Minkowski
- 3.10. Proof of Minkowski's theorem
- 3.11. Developments of Theorem $37$
- $\text {IV}$. IRRATIONAL NUMBERS
- 4.1. Some generalities
- 4.2. Numbers known to be irrational
- 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 $\sqrt 5$
- 4.7. Some more irrational numbers
- $\text {V}$. CONGRUENCES AND RESIDUES
- 5.1. Highest common divisor and least common multiple
- 5.2. Congruences and classes of residues
- 5.3. Elementary properties of congruences
- 5.4. Linear congruences
- 5.5. Euler's function $\map \phi m$
- 5.6. Applications of Theorems $59$ and $61$ to trigonometrical series
- 5.7. A general principle
- 5.8. Construction of the regular polygon of $17$ sides
- $\text {VI}$. FERMAT'S THEOREM AND ITS CONSEQUENCES
- 6.1. Fermat's theorem
- 6.2. Some properties of binomial coefficients
- 6.3. A second proof of Theorem $72$
- 6.4. Proof of Theorem $22$
- 6.5. Quadratic residues
- 6.6. Special cases of Theorem $79$: Wilson's theorem
- 6.7. Elementary properties of quadratic residues and non-residues
- 6.8. The order of $a \pmod m$
- 6.9. The converse of Fermat's theorem
- 6.10. Divisibility of $2^{p - 1} - 1$ by $p^2$
- 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
- $\text {VII}$. GENERAL PROPERTIES OF CONGRUENCES
- 7.1. Roots of congruences
- 7.2. Integral polynomials and identical congruences
- 7.3. Divisibility of polynomials (mod $m$)
- 7.4. Roots of congruences to a prime modulus
- 7.5. Some applications of the general theorems
- 7.6. Lagrange's proof of Fermst's and Wilson's theorems
- 7.7. The residue of $\set {\tfrac 1 2 \paren {p - 1} }!$
- 7.8. A theorem of Wolstenholme
- 7.9. The theorem of von Staudt
- 7.10. Proof of von Staudt's theorem
- $\text {VIII}$. CONGRUENCES TO COMPOSITE MODULI
- 8.1. Linear congruences
- 8.2. Congruences of higher degree
- 8.3. Congruences to a prime-power modulus
- 8.4. Examples
- 8.5. Bauer's identical congruence
- 8.6. Bauer's congruence: the case $p = 2$
- 8.7. A theorem of Leudesdorf
- 8.8. Further consequences of Bauer's theorem
- 8.9. The residues of $2^{p - 1}$ and $\paren {p - 1}!$ to modulus $p^2$
- $\text {IX}$. 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 numbers 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
- $\text {X}$. CONTINUED FRACTIONS
- 10.1. Finite continued fractions
- 10.2. Convergents to a continued fraction
- 10.3. Continued fractions with positive quotients
- 10.4. Simple continued 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 convergence
- 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. Periodic continued fractions
- 10.13. Some special quadratic surds
- 10.14. The series of Fibonacci and Lucas
- 10.15. Approximation by convergents
- $\text {XI}$. APPROXIMATION OF IRRATIONALS BY RATIONALS
- 11.1. Statement of the problem
- 11.2. Generalities concerning the problem
- 11.3. An argument of Dirichlet
- 11.4. Orders of approximation
- 11.5. Algebraic and transcendental members
- 11.6. The existence of transcendental 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 concerning approximation
- 11.12. Simultaneous approximation
- 11.13. The transcendence of $e$
- 11.14. The transcendence of $\pi$
- $\text {XII}$. THE FUNDAMENTAL THEOREM OF ARITHMETIC IN $\map k 1$, $\map k i$, AND $\map k \rho$
- 12.1. Algebraic numbers and integers
- 12.2. The rational integers, the Gaussian integers, and the integers of $\map k \rho$
- 12.3. Euclid's algorithm
- 12.4. Application of Euclid's algorithm to the fundamental theorem in $\map k 1$
- 12.5. Historical remarks on Euclid's algorithm and the fundamental theorem
- 12.6. Properties of the Gaussian integers
- 12.7. Primes in $\map k i$
- 12.8. The fundamental theorem of arithmetic in $\map k i$
- 12.9. The integers of $\map k \rho$
- $\text {XIII}$. SOME DIOPHANTINE EQUATIONS
- 13.1. Fermat's last theorem
- 13.2. The equation $x^2 + y^2 = z^2$
- 13.3. The equation $x^4 + y^4 = z^4$
- 13.4. The equation $x^3 + y^3 = z^3$
- 13.5. The equation of $x^3 + y^3 = 3z^3$
- 13.6. The expression of a rations as a sum of rational cubes
- 13.7. The equation $x^3 + y^3 + x^3 = t^3$
- $\text {XIV}$. QUADRATIC FIELDS (1)
- 14.1. Algebraic melds
- 14.2. Algebraic numbers and integers; primitive polynomials
- 14.3. The general quadratic field $\map k {\sqrt m}$
- 14.4. Unities and primes
- 14.5. The unities of $\map k {\sqrt 2}$
- 14.6. Fields in which the fundamental theorem is false
- 14.7. Complex Euclidean fields
- 14.8. Real Euclidean fields
- 14.9. Real Euclidean fields (continued)
- $\text {XV}$. QUADRATIC FIELDS (2)
- 15.1. The primes of $\map k i$
- 15.2. Fermat's theorem in $\map k i$
- 15.3. The primes of $\map k \rho$
- 15.4. The primes of $\map k {\sqrt 2}$ and $\map k {\sqrt 5}$
- 15.5. Lucas's test for the primarily of the Mersenne number $M_{4 n + 3}$
- 15.6. General remarks on the arithmetic of quadratic fields
- 15.7. Ideals in a quadratic fields
- 15.8. Other fields
- $\text {XVI}$. THE ARITHMETICAL FUNCTIONS $\map \phi n$, $\map d n$, $\map \sigma n$, $\map r n$
- 16.1. The function $\map \phi n$
- 16.2. A further proof of Theorem $63$
- 16.3. The Möbius function
- 16.4. The Möbius inversion formula
- 16.5. Further inversion formulae
- 16.6. Evaluation of Ramanujan's sum
- 16.7. The functions $\map d n$ and $\map {\sigma_k} n$
- 16.8. Perfect numbers
- 16.9. The function $\map r n$
- 16.10. Proof of the formula for $\map r n$
- $\text {XVII}$. GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS
- 17.1 The generation of arithmetical functions by means of Dirichlet series
- 17.2. The zeta function
- 17.3. The behaviour of $\map \zeta s$ when $s \to 1$
- 17.4. Multiplication of Dirichlet series
- 17.5. The generating functions of some special arithmetical functions
- 17.6. The analytical interpretation of the Möbius formula
- 17.7. The function $\map \Lambda n$
- 17.8. Further examples of generating functions
- 17.9. The generating function of $\map r n$
- 17.10. Generating functions of other types
- $\text {XVIII}$. THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS
- 18.1. The order of $\map d n$
- 18.2. The average order of $\map d n$
- 18.3. The order of $\map \sigma n$
- 18.4. The order of $\map \phi n$
- 18.5. The average order of $\map \phi n$
- 18.6. The number of squarefree numbers
- 18.7. The order of $\map r n$
- $\text {XIX}$. PARTITIONS
- 19.1. The general problem of additive arithmetic
- 19.2. Partitions of numbers
- 19.3. The incepting function of $\map p n$
- 19.4. Other generating functions
- 19.5. Two theorems of Euler
- 19.6. Further algebraical identities
- 19.7. Another formula for $\map F x$
- 19.8. A theorem of Jacobi
- 19.9. Special cases of Jacobi's identity
- 19.10. Applications of Theorem $353$
- 19.11. Elementary proof of Theorem $358$
- 19.12. Congruence properties of $\map p n$
- 19.13. The Rogers-Ramanujan identities
- 19.14. Proof of Theorems $362$ and $363$
- 19.15. Ramanujan's continued fraction
- $\text {XX}$. THE REPRESENTATION OF A NUMBER BY TWO OR FOUR SQUARES
- 20.1. Waring's problem: the numbers $\map g k$ and $\map G k$
- 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 $\map g 2$ and $\map G 2$
- 20.11. Lemma 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
- $\text {XXI}$. REPRESENTATION BY CUBES AND HIGHER POWERS
- 21.1. Biquadrates
- 21.2. Cubes: the existence of $\map G 3$ and $\map g 3$
- 21.3. A bound for $\map g 3$
- 21.4. Higher powers
- 21.5. A lower bound for $\map g k$
- 21.6. Lower bounds for $\map G k$
- 21.7. Sums affected with signs: the number $\map v k$
- 21.8. Upper bounds for $\map v k$
- 21.9. The problem of Prouhet and Tarry: the number $\map P {k, j}$
- 21.10. Evaluation of $\map P {k, j}$ for particular $k$ and $j$
- 21.11. Further problems of Diophantine analysis
- $\text {XXII}$. THE SERIES OF PRIMES (3)
- 22.1. The functions $\map \vartheta x$ and $\map \phi x$
- 22.2. Proof that $\map \vartheta x$ and $\map \phi x$ 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 sum $\sum p^{-1}$ and the product $\prod \paren {1 - p^{-1} }$
- 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 $\map \omega n$ and $\map \Omega n$
- 22.12. A note on round numbers
- 22.13. The normal order of $\map d n$
- 22.14. Selberg's theorem
- 22.15. The functions $\map R x$ and $\map V \xi$
- 22.16. Completion of the proof of theorems $434$, $6$ and $8$
- 22.17. Proof of Theorem $335$
- 22.18. Products of $k$ prime factors
- 22.19. Primes in an interval
- 22.20. A conjecture about the distribution of prime pairs $p$, $p + 2$
- $\text {XXIII}$. KRONECKER'S THEOREM
- 23.1. Kronecker'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
- $\text {XXIV}$. GEOMETRY OF NUMBERS
- 24.1. Introduction and restatement of the fundamental theorem
- 24.2. Simple applications
- 24.3. Arithmetical proof of Theorem 448
- 24.4. Best possible inequalities
- 24.5. The best possible inequality for $\xi^2 + \eta^2$
- 24.6. The best possible inequality for $\size {\xi \eta}$
- 24.7. A theorem concerning non-homogeneous forms
- 24.8. Arithmetical proof of Theorem $455$
- 24.9. Tchebotaref's theorem
- 24.10. A converse of Minkowski's Theorem $446$
- APPENDIX
- l. Another formula for $p_n$
- 2. A generalisation of Theorem $22$
- 3. Unsolved problems concerning primes
- A LIST OF BOOKS
- INDEX OF SPECIAL SYMBOLS AND WORDS
- INDEX OF NAMES
Further Editions
Click here for errata
Source work progress
- 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.4$ The sequence of primes