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

Subject Matter

 * Number Theory

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



Source work progress
* : Chapter $2$: Divisibility Theory in the Integers: $2.4$ The Diophantine Equation $a x + b y = c$


 * Exercises to be completed:


 * : Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm: Problems $2.3$: $4 \ \text {(a)}$
 * : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $5 \ \text {(b)}$