Contents |
Notation |
xi |
|
1 | Divisibility |
1 |
1.1 Introduction 1 | ||
1.2 Divisibility 4 | 1.3 Primes 20 | |
1.4 The Binomial Theorem 35 | ||
Notes on Chapter 1 44 | ||
2 | Congruences |
47 |
2.1 Congruences 47 | ||
2.2 Solutions of Congruences 60 | ||
2.3 The Chinese Remainder Theorem 64 | ||
2.4 Techniques of Numerical Calculation 74 | ||
2.5 Public-Key Cryptography 84 | ||
2.6 Prime Power Moduli 86 | ||
2.7 Prime Modulus 91 | ||
2.8 Primitive Roots and Power Residues 97 | ||
2.9 Congruences of Degree Two, Prime Modulus 110 | ||
2.10 Number Theory from an Algebraic Viewpoint 115 | ||
2.11 Groups, Rings, and Fields 121 | ||
Notes on Chapter 2 128 | ||
3 | Quadratic Reciprocity and Quadratic Forms |
131 |
3.1 Quadratic Residues 131 | ||
3.2 Quadratic Reciprocity 137 | ||
3.3 The Jacobi Symbol 142 | ||
3.4 Binary Quadratic Forms 150 | ||
3.5 Equivalence and Reduction of Binary Quadratic Forms 155 | ||
3.6 Sums of Two Squares 163 | ||
3.7 Positive Definite Binary Quadratic Forms 170 | ||
  Notes on Chapter 3 176 | ||
4 | Some Functions of Number Theory |
180 |
4.1 Greatest Integer Function 180 | 4.2 Arithmetic Functions 188 | 4.3 The Möbius Inversion Formula 193 | 4.4 Recurrence Functions 197 | 4.5 Combinatorial Number Theory 206 |
  Notes on Chapter 4 211 | ||
5 | Some Diophantine Equations |
212 |
5.1 The Equation ax + by = c 212 | ||
5.2 Simultaneous Linear Equations 219 | ||
5.3 Pythagorean Triangles 231 | ||
5.4 Assorted Examples 234 | ||
5.5 Ternary Quadratic Forms 240 | ||
5.6 Rational Points on Curves 249 | ||
5.7 Elliptic Curves 261 | ||
5.8 Factorization Using Elliptic Curves 281 | ||
5.9 Curves of Genus Greater Than 1 288 | ||
  Notes on Chapter 5 289 | ||
6 | Farey Fractions and Irrational Numbers |
297 |
6.1 Farey Sequences 297 | ||
6.2 Rational Approximation 301 | ||
6.3 Irrational Numbers 307 | ||
6.4 The Geometry of Numbers 312 | ||
  Notes on Chapter 6 322 | ||
7 | Simple Continued Fractions |
325 |
7.1 The Euclidean Algorithm 325 | ||
7.2 Uniqueness 327 | ||
7.3 Infinite Continued Fractions 329 | ||
7.4 Irrational Numbers 334 | ||
7.5 Approximation to Irrational Numbers 336 | ||
7.6 Best Possible Approximations 341 | ||
7.7 Periodic Continued Fractions 344 | ||
7.8 Pell's Equation 351 | ||
7.9 Numerical Computation 358 | ||
  Notes on Chapter 7 359 | ||
8 | Primes and Multiplicative Number Theory |
360 |
8.1 Elementary Prime Number Estimates 360 | ||
8.2 Dirichlet Series 374 | ||
8.3 Estimates of Arithmetic Functions 389 | ||
8.4 Primes in Arithmetic Progressions 401 | ||
  Notes on Chapter 8 406 | ||
9 | Algebraic Numbers |
409 |
9.1 Polynomials 410 | ||
9.2 Algebraic Numbers 414 | ||
9.3 Algebraic Number Fields 419 | ||
9.4 Algebraic Integers 424 | ||
9.5 Quadratic Fields 425 | ||
9.6 Units in Quadratic Fields 428 | ||
9.7 Primes in Quadratic Fields 429 | ||
9.8 Unique Factorization 431 | ||
9.9 Primes in Quadratic Fields Having the Unique Factoriation Property 433 | ||
9.10 The Equation x3 + y3 =  z3 441 | ||
  Notes on Chapter 9 445 | ||
10 | The Partition Function |
446 |
10.1 Partitions 446 | ||
10.2 Ferrers Graphs 448 | ||
10.3 Formal Power Series, Generating Functions, and Euler's Identity 452 | ||
10.4 Euler's Formula; Bounds on p(n) 497 | ||
10.5 Jacobi's Formula 463 | ||
10.6 A Divisibility Property 467 | ||
  Notes on Chapter 10 471 | ||
11 | The Density of Sequences of Integers |
472 |
11.1 Asymptotic Density 473 | ||
11.2 Schnirelmann Denstiy and the alpha-beta Theorem 476 |   Notes on Chapter 11 481 | |
Appendices |
482 |
|
A.1 The Fundamental Theorem of Algebra 482 | ||
A.2 Symmetric Functions 484 | ||
A.3 ASpecial Value of the Riemann Zeta Function 490 | ||
A.4 Linear Recurrences 493 | ||
General References |
500 |
|
Hints |
503 |
|
Answers |
512 |
|
Index |
522 |