# Elementary Number Theory McGraw-Hill Higher Education, 2007 - Number theory - 434 pages
Preface ix New to This Edition xi Preliminaries 01 (12) Mathematical Induction 01 (7) The Binomial Theorem 08 (5) Divisibility Theory in the Integers 13 (26) Early Number Theory 13 (4) The Division Algorithm 17 (2) The Greatest Common Divisor 19 (7) The Euclidean Algorithm 26 (6) The Diophantine Equation ax + by = c 32 (7) Primes and Their Distribution 39 (22) The Fundamental Theorem of Arithmetic 39 (5) The Sieve of Eratosthenes 44 (6) The Goldbach Conjecture 50 (11) The Theory of Congruences 61 (24) Carl Friedrich Gauss 61 (2) Basic Properties of Congruence 63 (6) Binary and Decimal Representations of Integers 69 (7) Linear Congruences and the Chinese Remainder Theorem 76 (9) Fermat's Theorem 85 (18) Pierre de Fermat 85 (2) Fermat's Little Theorem and Pseudoprimes 87 (6) Wilson's Theorem 93 (4) The Fermat-Kraitchik Factorization Method 97 (6) Number-Theoretic Functions 103 (26) The Sum and Number of Divisors 103 (9) The Mobius Inversion Formula 112 (5) The Greatest Integer Function 117 (5) An Application to the Calendar 122 (7) Euler's Generalization of Fermat's Theorem 129 (18) Leonhard Euler 129 (2) Euler's Phi-Function 131 (5) Euler's Theorem 136 (5) Some Properties of the Phi-Function 141 (6) Primitive Roots and Indices 147 (22) The Order of an Integer Modulo n 147 (5) Primitive Roots for Primes 152 (6) Composite Numbers Having Primitive Roots 158 (5) The Theory of Indices 163 (6) The Quadratic Reciprocity Law 169 (28) Euler's Criterion 169 (6) The Legendre Symbol and Its Properties 175 (10) Quadratic Reciprocity 185 (7) Quadratic Congruences with Composite Moduli 192 (5) Introduction to Cryptography 197 (20) From Caesar Cipher to Public Key Cryptography 197 (11) The Knapsack Cryptosystem 208 (5) An Application of Primitive Roots to Cryptography 213 (4) Numbers of Special Form 217 (28) Marin Mersenne 217 (2) Perfect Numbers 219 (6) Mersenne Primes and Amicable Numbers 225 (11) Fermat Numbers 236 (9) Certain Nonlinear Diophantine Equations 245 (16) The Equation x2 + y2 = z2 245 (7) Fermat's Last Theorem 252 (9) Representation of Integers as Sums of Squares 261 (22) Joseph Louis Lagrange 261 (2) Sums of Two Squares 263 (9) Sums of More Than Two Squares 272 (11) Fibonacci Numbers 283 (20) Fibonacci 283 (2) The Fibonacci Sequence 285 (7) Certain Identities Involving Fibonacci Numbers 292 (11) Continued Fractions 303 (46) Srinivasa Ramanujan 303 (3) Finite Continued Fractions 306 (13) Infinite Continued Fractions 319 (15) Pell's Equation 334 (15) Some Twentieth-Century Developments 349 (30) Hardy, Dickson, and Erdos 349 (5) Primality Testing and Factorization 354 (13) An Application to Factoring: Remote Coin Flipping 367 (4) The Prime Number Theorem and Zeta Function 371 (8) Miscellaneous Problems 379 (4) Appendixes 383 (38) General References 385 (4) Suggested Further Reading 389 (4) Tables 393 (16) Answers to Selected Problems 409 (12) Index 421.

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