Theory of Numbers

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)