Book:G.H. Hardy/An Introduction to the Theory of Numbers/Fifth Edition

Subject Matter

 * Number Theory

Contents

 * Preface to the Fifth Edition (Aberdeen, October 1978, )


 * Preface to the First Edition (Oxford, August 1938, and )


 * REMARKS ON NOTATION


 * 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


 * 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


 * 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


 * 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


 * 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 $\phi(m)$
 * 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


 * 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 $a (\bmod m)$
 * 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


 * 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 $\{\tfrac 1 2 (p - 1)\}!$
 * 7.8. A theorem of Wolstenholme
 * 7.9. The theorem of von Staudt
 * 7.10. Proof of von Staudt's theorem


 * 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 $(p-1)!$ to modulus $p^2$


 * 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


 * 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


 * 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 $\pi$


 * XII. THE FUNDAMENTAL THEOREM OF ARITHMETIC IN $k(1)$, $k(i)$, AND $k(\rho)$
 * 12.1. Algebraic numbers and integers
 * 12.2. The rational integers, the Gaussian integers, and the integers of $k(\rho)$
 * 12.3. Euclid's algorithm
 * 12.4. Application of Euclid's algorithm to the fundamental theorem in $k(1)$
 * 12.5. Historical remarks on Euclid's algorithm and the fundamental theorem
 * 12.6. Properties of the Gaussian integers
 * 12.7. Primes in $k(i)$
 * 12.8. The fundamental theorem of arithmetic in $k(i)$
 * 12.9. The integers of $k(\rho)$


 * 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 $x^3 + y^3 + x^3 = t^3$


 * XIV. QUADRATIC FIELDS (1)
 * 14.1. Algebraic melds
 * 14.2. Algebraic numbers and integers; primitive polynomials
 * 14.3. The general quadratic field $k(\sqrt m)$
 * 14.4. Unities and primes
 * 14.5. The unities of $k(\sqrt 2)$
 * 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)


 * XV. QUADRATIC FIELDS (2)
 * 15.1. The primes of $k(i)$
 * 15.2. Fermat's theorem in $k(i)$
 * 15.3. The primes of $k(\rho)$
 * 15.4. The primes of $k(\sqrt 2)$ and $k(\sqrt 5)$
 * 15.5. Lucas's test for the primarily of the Mersenne number $M_{4n+3}$
 * 15.6. General remarks on the arithmetic of quadratic fields
 * 15.7. Ideals in a quadratic fields
 * 15.8. Other fields


 * XVI. THE ARITHMETICAL FUNCTIONS $\phi(n)$, $d(n)$, $\sigma(n)$, $r(n)$
 * 16.1. The function $\phi(n)$
 * 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 $d(n)$ and $\sigma_k(n)$
 * 16.8. Perfect numbers
 * 16.9. The function $r(n)$
 * 16.10. Proof of the formula for $r(n)$


 * 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 $\zeta(s)$ 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 $\Lambda(n)$
 * 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 $\sigma(n)$
 * 18.4. The order of $\phi(n)$
 * 18.5. The average order of $\phi(n)$
 * 18.6. The number of squarefree numbers
 * 18.7. The order of $r(n)$


 * XIX. PARTITIONS
 * 19.1. The general problem of additive arithmetic
 * 19.2. Partitions of numbers
 * 19.3. The incepting function of $p(n)$
 * 19.4. Other generating functions
 * 19.5. Two theorems of Euler
 * 19.6. Further algebraical identities
 * 19.7. Another formula for $F(x)$
 * 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 $p(n)$
 * 19.13. The Rogers-Ramanujan identities
 * 19.14. Proof of Theorems 362 and 363
 * 19.15. Ramanujan's continued fraction


 * 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. 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


 * XXI. REPRESENTATION BY CUBES AND HIGHER POWERS
 * 21.1. Biquadrates
 * 21.2. Cubes: the existence 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 $G(k)$
 * 21.7. Sums affected 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 particular $k$ and $j$
 * 21.11. Further problems of Diophantine analysis


 * XXII. THE SERIES OF PRIMES (3)
 * 22.1. The functions $\vartheta(x)$ and $\phi(x)$
 * 22.2. Proof that $\vartheta(x)$ and $\phi(x)$ 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 $\prod (1-p^{-1})$
 * 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 $\omega(n)$ and $\Omega(n)$
 * 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 $V(\xi)$
 * 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 $p$, $p + 2$


 * 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


 * 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 $\left|{\xi \eta}\right|$
 * 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