Elementary Number Theory and Its Applications

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ISBN: 9780201578898 | 0201578891
Cover: Hardcover
Copyright: 1/1/1992

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Introduction	p. 1
The Integers
Basic properties	p. 4
Summations and products	p. 9
Mathematical induction	p. 15
Binomial coefficients	p. 28
Divisibility	p. 36
Representations of integers	p. 42
Computer operations with integers	p. 51
Complexity of integer operations	p. 57
Prime numbers	p. 64
Greatest Common Divisors and Prime Factorization
Greatest common divisors	p. 74
The Euclidean algorithm	p. 80
The fundamental theorem of arithmetic	p. 90
Fermat numbers and factorization methods	p. 103
Linear diophantine equations	p. 112
Congruences
Introduction to congruences	p. 119
Linear congruences	p. 131
The Chinese remainder theorem	p. 135
Systems of linear congruences	p. 145
Factoring using the Pollard rho method	p. 156
Applications of Congruences
Divisibility tests	p. 160
The perpetual calendar	p. 166
Round-robin tournaments	p. 171
Computer file storage and hashing functions	p. 173
Check digits	p. 178
Some Special Congruences
Wilson's theorem and Fermat's little theorem	p. 185
Pseudoprimes	p. 192
Euler's theorem	p. 201
Multiplicative Functions
Euler's phi-function	p. 207
The sum and number of divisors	p. 217
Perfect numbers and Mersenne primes	p. 223
Cryptology
Character ciphers	p. 234
Block ciphers	p. 245
Exponentiation ciphers	p. 253
Public-key cryptography	p. 259
Knapsack ciphers	p. 266
Some applications to computer science	p. 274
Primitive Roots
The order of an integer and primitive roots	p. 278
Primitive roots for primes	p. 285
Existence of primitive roots	p. 290
Index arithmetic	p. 298
Primality testing using primitive roots	p. 308
Universal exponents	p. 312
Pseudo-random numbers	p. 318
An application to the splicing of telephone cables	p. 324
Quadratic Residues and Reciprocity
Quadratic residues and nonresidues	p. 331
Quadratic reciprocity	p. 348
The Jacobi symbol	p. 357
Euler pseudoprimes	p. 367
Zero-knowledge proofs	p. 377
Decimal Fractions and Continued Fractions
Decimal fractions	p. 384
Finite continued fractions	p. 394
Infinite continued fractions	p. 405
Periodic continued fractions	p. 417
Factoring using continued fractions	p. 432
Some Nonlinear Diophantine Equations
Pythagorean triples	p. 436
Fermat's last theorem	p. 442
Sums of squares	p. 447
Pell's equation	p. 457
Appendix	p. 465
Answers to odd-numbered exercises	p. 481
Bibliography	p. 527
Index	p. 537
Table of Contents provided by Blackwell. All Rights Reserved.

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