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.
12. Prime Numbers.
13. Counting Primes.
14. Mersenne Primes.
15. Mersenne Primes and Perfect Numbers.
16. Powers Modulo m and Successive Squaring.
17. Computing kth Roots Modulo m.
18. Powers, Roots, and "Unbreakable" Codes.
19. Euler's Phi Function and Sums of Divisors.
20. Powers Modulo p and Primitive Roots.
21. Primitive Roots and Indices.
22. Squares Modulo p.
23. Is -1 a Square Modulo p? Is 2?
24. Quadratic Reciprocity.
25. Which Primes Are Sums of Two Squares?
26. Which Numbers Are Sums of Two Squares?
27. The Equation X4 + Y4 = Z4.
28. Square-Triangular Numbers Revisited.
29. Pell's Equation.
30. Diophantine Approximation.
31. Diophantine Approximation and Pell's Equation.
32. Primality Testing and Carmichael Numbers
33. Number Theory and Imaginary Numbers.
34. The Gaussian Integers and Unique Factorization.
35. Irrational Numbers and Transcendental Numbers.
36. Binomial Coefficients and Pascal's Triangle.
37. Fibonacci's Rabbits and Linear Recurrence Sequences.
38. Generating Functions.
39. Sums of Powers.
40. Cubic Curves and Elliptic Curves.
41. Elliptic Curves with Few Rational Points.
42. Points on Elliptic Curves Modulo p.
43. Torsion Collections Modulo p and Bad Primes.
44. Defect Bounds and Modularity Patterns.
45. Elliptic Curves and Fermat's Last Theorem.
Further Reading.
Appendix A: Factorization of Small Composite Integers.
Appendix B: List of Primes.
Index.
