Preface

Flowchart of Chapter Dependencies

Introduction

1. What Is Number Theory?

2. Pythagorean Triples

3. Pythagorean Triples and the Unit Circle

4. Sums of Higher Powers and Fermat’s Last Theorem

5. Divisibility and the Greatest Common Divisor

6. Linear Equations and the Greatest Common Divisor

7. Factorization and the Fundamental Theorem of Arithmetic

8. Congruences

9. Congruences, Powers, and Fermat’s Little Theorem

10. Congruences, Powers, and Euler’s Formula

11. Euler’s Phi Function and the Chinese Remainder Theorem

12. Prime Numbers

13. Counting Primes

14. Mersenne Primes

15. Mersenne Primes and Perfect Numbers

16. Powers Modulo *m* and Successive Squaring

17. Computing *k*^{th} Roots Modulo *m*

18. Powers, Roots, and “Unbreakable” Codes

19. Primality Testing and Carmichael Numbers

20. Squares Modulo p

21. Is -1 a Square Modulo *p*? Is 2?

22. Quadratic Reciprocity

23. Proof of Quadratic Reciprocity

24. Which Primes Are Sums of Two Squares?

25. Which Numbers Are Sums of Two Squares?

26. As Easy as One, Two, Three

27. Euler’s Phi Function and Sums of Divisors

28. Powers Modulo p and Primitive Roots

29. Primitive Roots and Indices

30. The Equation *X*^{4} + *Y*^{4} = *Z*^{4}

31. Square–Triangular Numbers Revisited

32. Pell’s Equation

33. Diophantine Approximation

34. Diophantine Approximation and Pell’s Equation

35. Number Theory and Imaginary Numbers

36. The Gaussian Integers and Unique Factorization

37. Irrational Numbers and Transcendental Numbers

38. Binomial Coefficients and Pascal’s Triangle

39. Fibonacci’s Rabbits and Linear Recurrence Sequences

40. Oh, What a Beautiful Function

41. Cubic Curves and Elliptic Curves

42. Elliptic Curves with Few Rational Points

43. Points on Elliptic Curves Modulo *p*

44. Torsion Collections Modulo *p* and Bad Primes

45. Defect Bounds and Modularity Patterns

46. Elliptic Curves and Fermat’s Last Theorem

Further Reading

Index

*47. The Topsy-Turvey World of Continued Fractions [online]

*48. Continued Fractions, Square Roots, and Pell’s Equation [online]

*49. Generating Functions [online]

*50. Sums of Powers [online]

*A. Factorization of Small Composite Integers [online]

*B. A List of Primes [online]

*These chapters are available online.