This introductory text is designed to entice non-math focused individuals into learning some mathematics, while teaching them to think mathematically. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results. Pythagorean Triples, Linear Equations and the Greatest Common Divisor, Factorization and the Fundamental Theorem of Arithmetic, Congruences, Mersenne Primes, Squares Modulo p, Quadratic Reciprocity, Pell's Equation, Diophantine Approximation, Irrational Numbers and Transcendental Numbers, Sums of Powers, Binomial Coefficients and Pascal's Triangle, Elliptic Curves and Fermat's Last Theorem. For individuals with limited math experience who are interested in number theory.
About the Author
Table of Contents1. 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.
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 k
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 X^4 + Y^4 = Z^4.
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 GaussianIntegers 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.
Appendix A: Factorization of Small Composite Integers.
Appendix B: List of Primes.