Table of Contents

Table of Contents

What is Number Theory?

The Integers.

• Numbers and Sequences.
• Sums and Products.
• Mathematical Induction.
• The Fibonacci Numbers.

Integer Representations and Operations.

• Representations of Integers.
• Computer Operations with Integers.
• Complexity of Integer Operations.

Primes and Greatest Common Divisors.

• Prime Numbers.
• The Distribution of Primes.
• Greatest Common Divisors.
• The Euclidean Algorithm.
• The Fundemental Theorem of Arithmetic.
• Factorization Methods and Fermat Numbers.
• Linear Diophantine Equations.

Congruences.

• Introduction to Congruences.
• Linear Congrences.
• The Chinese Remainder Theorem.
• Solving Polynomial Congruences.
• Systems of Linear Congruences.
• Factoring Using the Pollard Rho Method.

Applications of Congruences.

• Divisibility Tests.
• The perpetual Calendar.
• Round Robin Tournaments.
• Hashing Functions.
• Check Digits.

Some Special Congruences.

• Wilson's Theorem and Fermat's Little Theorem.
• Pseudoprimes.
• Euler's Theorem.

Multiplicative Functions.

• The Euler Phi-Function.
• The Sum and Number of Divisors.
• Perfect Numbers and Mersenne Primes.
• Mobius Inversion.
• Partitions.

Cryptology.

• Character Ciphers.
• Block and Stream Ciphers.
• Exponentiation Ciphers.
• Knapsack Ciphers.
• Cryptographic Protocols and Applications.

Primitive Roots.

• The Order of an Integer and Primitive Roots.
• Primitive Roots for Primes.
• The Existence of Primitive Roots.
• Index Arithmetic.
• Primality Tests Using Orders of Integers and Primitive Roots.
• Universal Exponents.

Applications of Primitive Roots and the Order of an Integer.

• Pseudorandom Numbers.
• The EIGamal Cryptosystem.
• An Application to the Splicing of Telephone Cables.

Quadratic Residues.

• Quadratic Residues and nonresidues.
• The Law of Quadratic Reciprocity.
• The Jacobi Symbol.
• Euler Pseudoprimes.
• Zero-Knowledge Proofs.

Decimal Fractions and Continued.

• Decimal Fractions.
• Finite Continued Fractions.
• Infinite Continued Fractions.
• Periodic Continued Fractions.
• Factoring Using Continued Fractions.

Some Nonlinear Diophantine Equations.

• Pythagorean Triples.
• Fermat's Last Theorem.
• Sums of Squares.
• Pell's Equation.
• Congruent Numbers.

The Gaussian Integers.

• Gaussian Primes.
• Unique Factorization of Gaussian Integers.
• Gaussian Integers and Sums of Squares.

About the Author

Kenneth H. Rosen received his BS in mathematics from the University of Michigan—Ann Arbor (1972) and his PhD in mathematics from MIT (1976). Before joining Bell Laboratories in 1982, he held positions at the University of Colorado—Boulder, The Ohio State University—Columbus, and the University of Maine—Orono, where he was an associate professor of mathematics. While working at AT&T Laboratories, he taught at Monmouth University, teaching courses in discrete mathematics, coding theory, and data security.

 

Dr. Rosen has published numerous articles in professional journals in the areas of number theory and mathematical modeling. He is the author of Elementary Number Theory, 6/e, and other books.