The Integers
- Numbers and Sequences
- Diophantine Approximation
- Sums and Products
- Mathematical Induction
- The Fibonacci Numbers
- Divisibility
Integer Representations and Operations
- Representations of Integers
- Computer Operations with Integers
- Complexity of Integer Operations
Greatest Common Divisors
- Greatest Common Divisors and Their Properties
- The Euclidean Algorithm
- Linear Diophantine Equations
Prime Numbers
- Prime Numbers
- The Distribution of Primes
- The Fundamental Theorem of Arithmetic
- Factorization Methods and the Fermat Numbers
Congruences
- Introduction to Congruences
- Linear Congruences
- The Chinese Remainder Theorem
- Polynomial Congruences
- Systems of Linear Congruences
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
Arithmetic Functions
- The Euler Phi-Function
- The Sum and Number of Divisors
- Perfect Numbers and Mersenne Primes
- Möbius Inversion
- Partitions
Cryptography
- Character Ciphers
- Block and Stream Ciphers
- Exponentiation Ciphers
- Public Key Cryptography
- Cryptographic Protocols and Applications
Primitive Roots
- The Order of an Integer and Primitive Roots
- Primitive Roots for Primes
- The Existence of Primitive Roots
- Discrete Logarithms and 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 Fractions
- Decimal Fractions
- Finite Continued Fractions
- Infinite Continued Fractions
- Periodic Continued Fractions
- Factoring Using Continued Fractions
Nonlinear Diophantine Equations and Elliptic Curves
- Pythagorean Triples
- Fermat's Last Theorem
- Sum of Squares
- Pell's Equation
- Congruent Numbers and Elliptic Curves
- Elliptic Curves Modulo Primes
- Applications of Elliptic Curves
The Gaussian Integers
- Gaussian Integers and Gaussian Primes
- Greatest Common Divisors and Unique Factorization
- Gaussian Integers and Sums of Squares

Elementary Number Theory, 7th edition

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