Book:David M. Burton/Elementary Number Theory/Revised Printing

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David M. Burton: Elementary Number Theory (Revised Edition)

Published $\text {1980}$, Allyn and Bacon

ISBN 0-205-06965-7


Subject Matter


Contents

Preface
Chapter 1. Some Preliminary Considerations
1.1 Mathematical Induction
1.2 The Binomial Theorem
1.3 Early Number Theory
Chapter 2. Divisibility Theory in the Integers
2.1 The Division Algorithm
2.2 The Greatest Common Divisor
2.3 The Euclidean Algorithm
2.4 The Diophantine Equation $a x + b y = c$
Chapter 3. Primes and Their Distribution
3.1 The Fundamental Theorem of Arithmetic
3.2 The Sieve of Eratosthenes
3.3 The Goldbach Conjecture
Chapter 4. The Theory of Congruences
4.1 Karl Friedrich Gauss
4.2 Basic Properties of Congruence
4.3 Special Divisibility Tests
4.4 Lienar Congruences
Chapter 5. Fermat's Theorem
5.1 Pierre de Fermat
5.2 Fermat's Factorization Method
5.3 The Little Theorem
5.4 Wilson's Theorem
Chapter 6. Number Theoretic Functions
6.1 The Functions $\tau$ and $\sigma$
6.2 The Möbius Inversion Formula
6.3 The Greatest Integer Function
Chapter 7. Euler's Generalization of Fermat's Theorem
7.1 Leonhard Euler
7.2 Euler's Phi-Function
7.3 Euler's Theorem
7.4 Some Properties of the Phi-Function
Chapter 8. Primitive Roots and Indices
8.1 The Order of an Integer Modulo $n$
8.2 Primitive Roots of Primes
8.3 Composite Numbers having Primitive Roots
8.4 The Theory of Indices
Chapter 9. The Quadratic Reciprocity Law
9.1 Euler's Criterion
9.2 The Legendre Symbol and its Properties
9.3 Quadratic Reciprocity
9.4 Quadratic Congruences with Composite Moduli
Chapter 10. Perfect Numbers
10.1 The Search for Perfect Numbers
10.2 Mersenne Primes
10.3 Fermat Numbers
Chapter 11. The Fermat Conjecture
11.1 Pythagorean Triples
11.2 The Famous "Last Theorem"
Chapter 12. Representation of Integers as Sums of Squares
12.1 Joseph Louis Lagrange
12.2 Sums of Two Squares
12.3 Sums of More than Two Squares
Chapter 13. Fibonacci Numbers and Continued Fractions
13.1 The Fibonacci Sequence
13.2 Certain Identities Involving Fibonacci Numbers
13.3 Finite Continued Fractions
13.4 Infinite Continued Fractions
13.5 Pell's Equation
Appendixes.
The Prime Number Theorem
References
Suggestions for Further Reading
Tables
Answers to Selected Problems
Index


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