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The Higher Arithmetic, Eighth Edition

Book Description

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.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Introduction
  6. I Factorization and the Primes
    1. 1. The laws of arithmetic
    2. 2. Proof by induction
    3. 3. Prime numbers
    4. 4. The fundamental theorem of arithmetic
    5. 5. Consequences of the fundamental theorem
    6. 6. Euclid’s algorithm
    7. 7. Another proof of the fundamental theorem
    8. 8. A property of the H.C.F
    9. 9. Factorizing a number
    10. 10. The series of primes
  7. II Congruences
    1. 1. The congruence notation
    2. 2. Linear congruences
    3. 3. Fermat’s theorem
    4. 4. Euler’s function 0(m)
    5. 5. Wilson’s theorem
    6. 6. Algebraic congruences
    7. 7. Congruences to a prime modulus
    8. 8. Congruences in several unknowns
    9. 9. Congruences covering all numbers
  8. III Quadratic Residues
    1. 1. Primitive roots
    2. 2. Indices
    3. 3. Quadratic residues
    4. 4. Gauss’s lemma
    5. 5. The law of reciprocity
    6. 6. The distribution of the quadratic residues
  9. IV Continued Fractions
    1. 1. Introduction
    2. 2. The general continued fraction
    3. 3. Euler’s rule
    4. 4. The convergents to a continued fraction Let
    5. 5. The equation ax — by = 1
    6. 6. Infinite continued fractions
    7. 7. Diophantine approximation
    8. 8. Quadratic irrationals
    9. 9. Purely periodic continued fractions
    10. 10. Lagrange’s theorem
    11. 11. Pell’s equation This is the equation
    12. 12. A geometrical interpretation of continued fractions
  10. V Sums of Squares
    1. 1. Numbers representable by two squares
    2. 2. Primes of the form 4k + 1
    3. 3. Constructions for x and y
    4. 4. Representation by four squares
    5. 5. Representation by three squares
  11. VI Quadratic Forms
    1. 1. Introduction
    2. 2. Equivalent forms
    3. 3. The discriminant
    4. 4. The representation of a number by a form
    5. 5. Three examples
    6. 6. The reduction of positive definite forms
    7. 7. The reduced forms
    8. 8. The number of representations
    9. 9. The class-number
  12. VII Some Diophantine Equations
    1. 1. Introduction
    2. 2. The equation x2+ y2 = z2
    3. 3. The equation ax2 + by2 = z2
    4. 4. Elliptic equations and curves
    5. 5. Elliptic equations modulo primes
    6. 6. Fermat’s Last Theorem
    7. 7. The equation x3 + y3 = z3 + w3
    8. 8. Further developments
  13. VIII Computers and Number Theory
    1. 1. Introduction
    2. 2. Testing for primality
    3. 3. 'Random' number generators
    4. 4. Pollard’s factoring methods
    5. 5. Factoring and primality via elliptic curves
    6. 6. Factoring large numbers
    7. 7. The Diffie-Hellman cryptographic method
    8. 8. The RSA cryptographic method
    9. 9. Primality testing revisited
  14. Exercises
  15. Hints
  16. Answers
  17. Bibliography
  18. Index