Elementary Number TheoryElementary Number Theory takes an accessible approach to teaching students about the role of number theory in pure mathematics and its important applications to cryptography and other areas. The first chapter of the book explains how to do proofs and includes a brief discussion of lemmas, propositions, theorems, and corollaries. The core of the tex |
Contents
| 1 | |
Chapter 1 Divisibility | 9 |
Chapter 2 Linear Diophantine Equations | 55 |
Chapter 3 Unique Factorization | 67 |
Chapter 4 Applications of Unique Factorization | 77 |
Chapter 5 Congruences | 95 |
Chapter 6 Fermat Euler Wilson | 133 |
Chapter 7 Cryptographic Applications | 153 |
Chapter 10 Quadratic Reciprocity | 245 |
Chapter 11 Primality and Factorization | 275 |
Chapter 12 Sums of Squares | 297 |
Chapter 13 Arithmetic Functions | 311 |
Chapter 14 Continued Fractions | 327 |
Chapter 15 Recent Developments | 341 |
Appendix A Supplementary Topics | 351 |
Appendix B Answers and Hints for OddNumbered Exercises | 377 |
Chapter 8 Order and Primitive Roots | 199 |
Chapter 9 More Cryptographic Applications | 229 |
Back Cover | 389 |
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Common terms and phrases
Alice and Bob Alice chooses assume calculate cards Chapter CHECK YOUR UNDERSTANDING Chinese Remainder Theorem ciphertext coefficients columns common divisor composite Compute congruence conjectured continued fraction Corollary cryptographic cryptosystem cube decimal expansion decryption divide encryption encryption exponent equation example Extended Euclidean Algorithm Fermat Primality Test Fermat's theorem formula gcd(a gcd(b gcd(m Hill cipher Hint implies induction infinitely many primes irrational Lemma look matrix Mersenne primes method mod 9 modulus multiplicative inverse nonnegative nonzero number of primes number theory obtain odd number odd prime pairs perfect numbers plaintext polynomial positive integer prime factors prime numbers primitive root primitive root mod problem product of primes proof Proposition prove Pythagorean triple Quadratic Reciprocity rational number rational roots real number relatively prime says secret Section sends Show solutions mod solve square mod statement Suppose true values write yields