-
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 kth Roots Modulo m
-
18. Powers, Roots, and “Unbreakable” Codes
-
19. Primality Testing and Carmichael Numbers
-
20. Squares Modulo p
-
21. Quadratic Reciprocity
-
22. Proof of Quadratic Reciprocity
-
23. Which Primes Are Sums of Two Squares?
-
24.Which Numbers Are Sums of Two Squares?
-
25. Euler’s Phi Function and Sums of Divisors
-
26. Powers Modulo p and Primitive Roots
-
27. Primitive Roots and Indices
-
28. The Equation X4 + Y4 = Z4
-
29. Square–Triangular Numbers Revisited
-
30. Pell’s Equation
-
31. Diophantine Approximation
-
32. Diophantine Approximation and Pell’s Equation
-
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. Cubic Curves and Elliptic Curves
-
39. Elliptic Curves with Few Rational Points
-
40. Points on Elliptic Curves Modulo p
-
41. Torsion Collections Modulo p and Bad Primes
-
42. Defect Bounds and Modularity Patterns
-
43. Elliptic Curves and Fermat’s Last Theorem
-
44. The Topsy-Turvy World of Continud Fractions
-
45. Continued Fractions and Pell’s Equation
-
46. Generating Functions
-
47. Sums of Powers
-
48. Appendix: A List of Primes
