Lecture Notes

These notes were prepared by Joseph Lee, a student in the class, in collaboration with Prof. Kumar.

SES # TOPICS
1 Introduction, Diophantine Equations, Divisibility, GCD (PDF)
2 Euclidean Algorithm, Primes (PDF)
3 Binomial Coefficients, Congruences (PDF)
4 FFermat, Euler, Wilson, Linear Congruences (PDF)
5 Linear Congruences, Chinese Remainder Theorem, Algorithms (PDF)
6 Primality, Factoring, RSA, Hensel's Lemma (PDF)
7 Congruences mod Primes, Order, Primitive Roots (PDF)
8 Primitive Roots (Prime Powers), Index Calculus (PDF)
9 Midterm 1
10 Quadratic Residues, Quadratic Reciprocity (PDF)
11 Jacobi Symbol, Computation, Zolotareff's Definition (PDF)
12 Square Roots, Tonelli's Algorithm, Number of Consecutive Pairs of Squares mod p (PDF)
13 Cyclotomic Polynomials, Primes Congruent to 1 mod n (PDF)
14 Arithmetic Functions (PDF)
15 Mobius Inversion Formula, Zeta Functions (PDF)
16 Linear Recurrences (PDF)
17 Generating Functions (PDF)
18 Midterm 2
19 More on Generating Functions, Two Squares Theorem (PDF)
20 Continued Fractions I (PDF)
21 Continued Fractions II: Inequalities (PDF)
22 Periodic Continued Fractions, Quadratic Irrationalities (PDF)
23 Brahmagupta-Pell Equation (PDF)
24 Four Squares Theorem (PDF)
25 Pythagorean Triples, Fermat Descent (PDF)
26 Rational Points on Conics (PDF)