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Preface | p. viii |
New to This Edition | p. x |
Preliminaries | p. 01 |
Mathematical Induction | p. 01 |
The Binomial Theorem | p. 08 |
Divisibility Theory in the Integers | p. 13 |
Early Number Theory | p. 13 |
The Division Algorithm | p. 17 |
The Greatest Common Divisor | p. 19 |
The Euclidean Algorithm | p. 26 |
The Diophantine Equation ax + by = c | p. 32 |
Primes and Their Distribution | p. 39 |
The Fundamental Theorem of Arithmetic | p. 39 |
The Sieve of Eratosthenes | p. 44 |
The Goldbach Conjecture | p. 50 |
The Theory of Congruences | p. 61 |
Carl Friedrich Gauss | p. 61 |
Basic Properties of Congruence | p. 63 |
Binary and Decimal Representations of Integers | p. 69 |
Linear Congruences and the Chinese Remainder Theorem | p. 76 |
Fermat's Theorem | p. 85 |
Pierre de Fermat | p. 85 |
Fermat's Little Theorem and Pseudoprimes | p. 87 |
Wilson's Theorem | p. 93 |
The Fermat-Kraitchik Factorization Method | p. 97 |
Number-Theoretic Functions | p. 103 |
The Sum and Number of Divisors | p. 103 |
The Möbius Inversion Formula | p. 112 |
The Greatest Integer Function | p. 117 |
An Application to the Calendar | p. 122 |
Euler's Generalization of Fermat's Theorem | p. 129 |
Leonhard Euler | p. 129 |
Euler's Phi-Function | p. 131 |
Euler's Theorem | p. 136 |
Some Properties of the Phi-Function | p. 141 |
Primitive Roots and Indices | p. 147 |
The Order of an Integer Modulo n | p. 147 |
Primitive Roots for Primes | p. 152 |
Composite Numbers Having Primitive Roots | p. 158 |
The Theory of Indices | p. 163 |
The Quadratic Reciprocity Law | p. 169 |
Euler's Criterion | p. 169 |
The Legendre Symbol and Its Properties | p. 175 |
Quadratic Reciprocity | p. 185 |
Quadratic Congruences with Composite Moduli | p. 192 |
Introduction to Cryptography | p. 197 |
From Caesar Cipher to Public Key Cryptography | p. 197 |
The Knapsack Cryptosystem | p. 209 |
An Application of Primitive Roots to Cryptography | p. 214 |
Numbers of Special Form | p. 219 |
Marin Mersenne | p. 219 |
Perfect Numbers | p. 221 |
Mersenne Primes and Amicable Numbers | p. 227 |
Fermat Numbers | p. 237 |
Certain Nonlinear Diophantine Equations | p. 245 |
The Equation x^{2} + y^{2} = z^{2} | p. 245 |
Fermat's Last Theorem | p. 252 |
Representation of Integers as Sums of Squares | p. 261 |
Joseph Louis Lagrange | p. 261 |
Sums of Two Squares | p. 263 |
Sums of More Than Two Squares | p. 272 |
Fibonacci Numbers | p. 283 |
Fibonacci | p. 283 |
The Fibonacci Sequence | p. 285 |
Certain Identities Involving Fibonacci Numbers | p. 292 |
Continued Fractions | p. 303 |
Srinivasa Ramanujan | p. 303 |
Finite Continued Fractions | p. 306 |
Infinite Continued Fractions | p. 319 |
Farey Fractions | p. 334 |
Pell's Equation | p. 337 |
Some Modern Developments | p. 353 |
Hardy, Dickson, and Erdös | p. 353 |
Primality Testing and Factorization | p. 358 |
An Application to Factoring: Remote Coin Flipping | p. 371 |
The Prime Number Theorem and Zeta Function | p. 375 |
Miscellaneous Problems | p. 384 |
Appendixes | p. 387 |
General References | p. 387 |
Suggested Further Reading | p. 390 |
Tables | p. 393 |
Answers to Selected Problems | p. 410 |
Index | p. 421 |
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