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Introduction 1

1 What Is Number Theory? 6

2 Pythagorean Triples 13

3 Pythagorean Triples and the Unit Circle 20

4 Sums of Higher Powers and Fermat's Last Theorem 24

5 Divisibility and the Greatest Common Divisor 28

6 Linear Equations and the Greatest Common Divisor 35

7 Factorization and the Fundamental Theorem of Arithmetic 44

8 Congruences 53

9 Congruences,Powers,and Fermat's Little Theorem 60

10 Congruences,Powers,and Euler's Formula 66

11 Euler's Phi Function and the Chinese Remainder Theorem 70

12 Prime Numbers 78

13 Counting Primes 85

14 Mersenne Primes 91

15 Mersenne Primes and Perfect Numbers 95

16 Powers Modulo m and Successive Squaring 105

17 Computing kth Roots Modulo m 112

18 Powers,Roots,and"Unbreakable"Codes 117

19 Primality Testing and Carmichael Numbers 123

20 Euler's Phi Function and Sums of Divisors 134

21 Powers Modulo p and Primitive Roots 139

22 Primitive Roots and Indices 149

23 Squares Modulo p 156

24 Is-1 a Square Modulo p?Is 2? 164

25 Quadratic Reciprocity 175

26 Which Primes Are Sums of Two Squares? 186

27 Which Numbers Are Sums of Two Squares? 198

28 The Equation X4+Y4=Z4 204

29 Square-Triangular Numbers Revisited 207

30 Pell's Equation 216

31 Diophantine Approximation 222

32 Diophantine Approximation and Pell's Equation 232

33 Number Theory and Imaginary Numbers 239

34 The Gaussian Integers and Unique Factorization 253

35 Irrational Numbers and Transcendental Numbers 270

36 Binomial Coefficients and Pascal's Triangle 286

37 Fibonacci's Rabbits and Linear Recurrence Sequences 297

38 Oh,What a Beautiful Function 310

39 The Topsy-Turvy World of Continued Fractions 324

40 Continued Fractions,Square Roots,and Pell's Equation 340

41 Generating Functions 355

42 Sums of Powers 365

43 Cubic Curves and Elliptic Curves 376

44 Elliptic Curves with Few Rational Points 388

45 Points on Elliptic Curves Modulo p 395

46 Torsion Collections Modulo p and Bad Primes 406

47 Defect Bounds and Modularity Patterns 410

48 Elliptic Curves and Fermat's Last Theorem 416

Further Reading 418

A Factorization of Small Composite Integers 419

B A List of Primes 421

Index 423