What is Number Theory? Pythagorean Triples. Pythagorean Triples and the Unit Circle. Sums of Higher Powers and Fermat's Last Theorem. Divisibility and the Greatest Common Divisor. Linear Equations and the Greatest Common Divisor. Factorization and the Fundamental Theorem of Arithmetic. Congruences. Congruences, Powers, and Fermat's Little Theorem. Congruences, Powers, and Euler's Formula. Euler's Phi Function. Prime Numbers. Counting Primes. Mersenne Primes. Mersenne Primes and Perfect Numbers. Powers Modulo m and Successive Squaring. Computing kth Roots Modulo m. Powers, Roots, and "Unbreakable" Codes. Euler's Phi Function and Sums of Divisors. Powers Modulo p and Primitive Roots. Primitive Roots and Indices. Squares Modulo p. Is -1 a Square Modulo p? Is 2? Quadratic Reciprocity. Which Primes are Sums of Two Squares. Which Numbers are Sums of Two Squares. The Equation X4 + Y4 = Z4. Square-Triangular Numbers Revisited. Pell's Equation. Diophantine Approximation. Diophantine Approximation and Pell's Equation. Cubic Curves and Elliptic Curves. Elliptic Curves with Few Rational Points. Points on Elliptic Curves Modulo p. Torsion Collections Modulo p and Bad Primes. Defect Bounds and Modularity Patterns. Elliptic Curves and Fermat's Last Theorem. Appendix A: Factorization of Small Composite Integers. Appendix B: List of Primes. Additional Exercises.
