Preface | v | ||||
Introduction | 1 | (65) | |||
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6 | (7) | |||
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13 | (7) | |||
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20 | (4) | |||
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24 | (4) | |||
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28 | (7) | |||
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35 | (9) | |||
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44 | (9) | |||
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53 | (7) | |||
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60 | (6) | |||
10 Congruences, Powers, and Euler's Formula | 66 | (4) | |||
11 Euler's Phi Function and the Chinese Remainder Theorem | 70 | (8) | |||
12 Prime Numbers | 78 | (7) | |||
13 Counting Primes | 85 | (6) | |||
14 Mersenne Primes | 91 | (4) | |||
15 Mersenne Primes and Perfect Numbers | 95 | (10) | |||
16 Powers Modulo m and Successive Squaring | 105 | (7) | |||
17 Computing kth Roots Modulo m | 112 | (5) | |||
18 Powers, Roots, and "Unbreakable" Codes | 117 | (6) | |||
19 Primality Testing and Carmichael Numbers | 123 | (11) | |||
20 Euler's Phi Function and Sums of Divisors | 134 | (5) | |||
21 Powers Modulo p and Primitive Roots | 139 | (10) | |||
22 Primitive Roots and Indices | 149 | (7) | |||
23 Squares Modulo p | 156 | (8) | |||
24 Is -1 a Square Modulo p? Is 2? | 164 | (11) | |||
25 Quadratic Reciprocity | 175 | (11) | |||
26 Which Primes Are Sums of Two Squares? | 186 | (12) | |||
27 Which Numbers Are Sums of Two Squares? | 198 | (6) | |||
28 The Equation X4 + Y4 = Z4 | 204 | (3) | |||
29 Square-Triangular Numbers Revisited | 207 | (9) | |||
30 Pell's Equation | 216 | (6) | |||
31 Diophantine Approximation | 222 | (10) | |||
32 Diophantine Approximation and Pell's Equation | 232 | (7) | |||
33 Number Theory and Imaginary Numbers | 239 | (14) | |||
34 The Gaussian Integers and Unique Factorization | 253 | (17) | |||
35 Irrational Numbers and Transcendental Numbers | 270 | (16) | |||
36 Binomial Coefficients and Pascal's Triangle | 286 | (11) | |||
37 Fibonacci's Rabbits and Linear Recurrence Sequences | 297 | (13) | |||
38 Oh, What a Beautiful Function | 310 | (14) | |||
39 The Topsy-Turvy World of Continued Fractions | 324 | (16) | |||
40 Continued Fractions, Square Roots, and Pell's Equation | 340 | (15) | |||
41 Generating Functions | 355 | (10) | |||
42 Sums of Powers | 365 | (11) | |||
43 Cubic Curves and Elliptic Curves | 376 | (12) | |||
44 Elliptic Curves with Few Rational Points | 388 | (7) | |||
45 Points on Elliptic Curves Modulo p | 395 | (11) | |||
46 Torsion Collections Modulo p and Bad Primes | 406 | (4) | |||
47 Defect Bounds and Modularity Patterns | 410 | (6) | |||
48 Elliptic Curves and Fermat's Last Theorem | 416 | (2) | |||
Further Reading | 418 | (1) | |||
A Factorization of Small Composite Integers | 419 | (2) | |||
B A List of Primes | 421 | (2) | |||
Index | 423 |
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