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list of symbols x what is number theory? 1 the integers 5 1.1 numbers and sequences 5 1.2 sums and products 16 1.3 mathematical induction 23 1.4 the fibonacci numbers 30 1.5 divisibility 36 2 integer representations and operations 45 2.1 representations of integers 45 2.2 computer operations with integers 54 2.3 complexity of integer operations 61 3 primes and greatest common divisors 69 3.1 prime numbers 70 3.2 the distribution of primes 79 3.3 greatest common divisors and their properties 93 3.4 the euclidean algorithm 102 3.5 the fundamental theorem of arithmetic 112 3.6 factorization methods and the fermat numbers 127 3.7 linear diophantine equations 137 4 congruences 145 4.1 introduction to congruences 145 4.2 linear congruences 157 4.3 the chinese remainder theorem 162 4.4 solving polynomial congruences 171 4.5 systems of linear congruences 178 4.6 factoring using the pollard rho method 187 5 applications of congruences 191 5.1 divisibility tests 191 5.2 the perpetual calendar 197 5.3 round-robin tournaments 202 5.4 hashing functions 204 5.5 check digits 209 6 some special congruences 217 6.1 wilson's theorem and fermat's little theorem 217 6.2 pseudoprimes 225 6.3 euler's theorem 234 7 multiplicative functions 239 7.1 the euler phi-function 239 7.2 the sum and number of divisors 249 7.3 perfect numbers and mersenne primes 256 7.4 misbius inversion 269 7.5 partitions 277 8 cryptology 291 8.1 character ciphers 291 8.2 block and stream ciphers 300 8.3 exponentiation ciphers 318 8.4 public key cryptography 321 8.5 knapsack ciphers 331 8.6 cryptographic protocols and applications 338 9 primitive roots 347 9.1 the order of an integer and primitive roots 347 9.2 primitive roots for primes 354 9.3 the existence of primitive roots 360 9.4 discrete logarithms and index arithmetic 368 9.5 primality tests using orders of integers and primitive roots 378 9.6 universal exponents 385 10 applications of primitive roots and the order of an integer 393 10.1 pseudorandom numbers 393 10.2 the eigamal cryptosystem 402 10.3 an application to the splicing of telephone cables 408 11 quadratic residues 415 11.1 quadratic residues and nonresidues 416 11.2 the law of quadratic reciprocity 430 11.3 the jacobi symbol 443 11.4 euler pseudoprimes 453 11.5 zero-knowledge proofs 461 12 decimal fractions and continued fractions 469 12.1 decimal fractions 469 12.2 finite continued fractions 481 12.3 infinite continued fractions 491 12.4 periodic continued fractions 503 12.5 factoring using continued fractions 517 13 some nonlinear diophantine equations 521 13.1 pythagorean triples 522 13.2 fermat's last theorem 530 13.3 sums of squares 542 13.4 pell's equation 553 13.5 congruent numbers 560 14 the gaussian integers 577 14.1 gaussian integers and gaussian primes 577 14.2 greatest common divisors and unique factorization 589 14.3 gaussian integers and sums of squares 599 appendix a axioms for the set of integers 605 appendix b binomial coefficients 608 appendix c using maple and mathematica for number theory 615 c.1 using maple for number theory 615 c.2 using mathematica for number theory 619 appendix d number theory web links 624 appendix e tables 626 answers to odd-numbered exercises 641 bibliography 721 index of biographies 733 index 735 photo credits 752