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The Higher Arithmetic An Introduction to the Theory of Numbers 8th edition by Davenport ISBN 9781139636551 1139636553

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ISBN 10: 1139636553
ISBN 13: 9781139636551
Author: H. Davenport

The Higher Arithmetic
An Introduction to the Theory of Numbers

The theory of numbers is generally considered to be the ‘purest’ branch of pure mathematics and demands exactness of thought and exposition from its devotees. It is also one of the most highly active and engaging areas of mathematics. Now into its eighth edition The Higher Arithmetic introduces the concepts and theorems of number theory in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical significance. Since earlier editions, additional material written by J. H. Davenport has been added, on topics such as Wiles’ proof of Fermat’s Last Theorem, computers and number theory, and primality testing. Written to be accessible to the general reader, with only high school mathematics as prerequisite, this classic book is also ideal for undergraduate courses on number theory, and covers all the necessary material clearly and succinctly.

The Higher Arithmetic An Introduction to the Theory of Numbers 8th Table of contents:

I. Factorization and the Primes

  1. The laws of arithmetic
  2. Proof by induction
  3. Prime numbers
  4. The fundamental theorem of arithmetic
  5. Consequences of the fundamental theorem
  6. Euclid’s algorithm
  7. Another proof of the fundamental theorem
  8. A property of the H.C.F
  9. Factorizing a number
  10. The series of primes

II. Congruences

  1. The congruence notation
  2. Linear congruences
  3. Fermat’s theorem
  4. Euler’s function 0(m)
  5. Wilson’s theorem
  6. Algebraic congruences
  7. Congruences to a prime modulus
  8. Congruences in several unknowns
  9. Congruences covering all numbers

III. Quadratic Residues

  1. Primitive roots
  2. Indices
  3. Quadratic residues
  4. Gauss’s lemma
  5. The law of reciprocity
  6. The distribution of the quadratic residues

IV. Continued Fractions

  1. Introduction
  2. The general continued fraction
  3. Euler’s rule
  4. The convergents to a continued fraction Let
  5. The equation ax — by = 1
  6. Infinite continued fractions
  7. Diophantine approximation
  8. Quadratic irrationals
  9. Purely periodic continued fractions
  10. Lagrange’s theorem
  11. Pell’s equation This is the equation
  12. A geometrical interpretation of continued fractions

V. Sums of Squares

  1. Numbers representable by two squares
  2. Primes of the form 4k + 1
  3. Constructions for x and y
  4. Representation by four squares
  5. Representation by three squares

VI. Quadratic Forms

  1. Introduction
  2. Equivalent forms
  3. The discriminant
  4. The representation of a number by a form
  5. Three examples
  6. The reduction of positive definite forms
  7. The reduced forms
  8. The number of representations
  9. The class-number

VII. Some Diophantine Equations

  1. Introduction
  2. The equation x² + y² = z²
  3. The equation ax² + by² = z²
  4. Elliptic equations and curves
  5. Elliptic equations modulo primes
  6. Fermat’s Last Theorem
  7. The equation x³ + y³ = z³ + w³
  8. Further developments

VIII. Computers and Number Theory

  1. Introduction
  2. Testing for primality
  3. ‘Random’ number generators
  4. Pollard’s factoring methods
  5. Factoring and primality via elliptic curves
  6. Factoring large numbers
  7. The Diffie-Hellman cryptographic method
  8. The RSA cryptographic method
  9. Primality testing revisited

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