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Credits | ||
Preface | ||
What Is Number Theory? | 1 | |
1 | Divisibility | 13 |
1.1 | The GCD and LCM | 13 |
1.2 | The Division Algorithm | 18 |
1.3 | The Euclidean Algorithm | 23 |
1.4 | Linear Combinations | 27 |
1.5 | Congruences | 33 |
1.6 | Mathematical Induction | 38 |
2 | Prime Numbers | 47 |
2.1 | Prime Factorization | 47 |
2.2 | The Fundamental Theorem of Arithmetic | 52 |
2.3 | The Importance of Unique Factorization | 55 |
2.4 | Prime Power Factorizations | 60 |
2.5 | The Set of Primes is Infinite | 65 |
2.6 | A Formula for [tau](n) | 71 |
3 | Numerical Functions | 77 |
3.1 | The Sum of the Divisors | 77 |
3.2 | Multiplicative Functions | 83 |
3.3 | Perfect Numbers | 90 |
3.4 | Mersenne and Fermat Numbers | 94 |
3.5 | The Euler Phi Function | 101 |
3.6 | The Mobius Inversion Formula | 108 |
4 | The Algebra of Congruence Classes | 117 |
4.1 | Solving Linear Congruences | 117 |
4.2 | The Clinese Remainder Theorem | 125 |
4.3 | The Theorems of Fermat and Euler | 133 |
4.4 | Primality Testing | 140 |
4.5 | Public-Key Cryptography | 148 |
5 | Congruences of Higher Degree | 159 |
5.1 | Polynomial Congruences | 159 |
5.2 | Congruences with Prime Power Moduli | 165 |
5.3 | Quadratic Residues | 172 |
5.4 | Quadratic Reciprocity | 179 |
5.5 | Flipping a Coin over the Telephone | 186 |
6 | The Number Theory of the Reals | 195 |
6.1 | Rational and Irrational Numbers | 195 |
6.2 | Finite Continued Fractions | 203 |
6.3 | Infinite Continued Fractions | 211 |
6.4 | Decimal Representation | 220 |
6.5 | Lagrange's Theorem and Primitive Roots | 226 |
7 | Diophantine Equations | 235 |
7.1 | Pythagorean Triples | 235 |
7.2 | Sums of Two Squares | 242 |
7.3 | Sums of Four Squares | 247 |
7.4 | Sums of Fourth Powers | 251 |
7.5 | Pell's Equation | 256 |
Bibliography | 263 | |
Answers to Odd-Numbered Problems | 267 | |
Index | 275 |
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Add Elementary number theory, This book is intended for an introductory course in number theory, primarily taken by mathematics majors. The author wrote the text with four main goals in mind. He wrote a book that is both easy to read and easy to teach from. The author also aims to eas, Elementary number theory to the inventory that you are selling on WonderClubX
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Add Elementary number theory, This book is intended for an introductory course in number theory, primarily taken by mathematics majors. The author wrote the text with four main goals in mind. He wrote a book that is both easy to read and easy to teach from. The author also aims to eas, Elementary number theory to your collection on WonderClub |