Math 11N: General Course Outline
Catalog Description
MATH 11N. Gateway to Mathematics: Number Theory. Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B. Introductory number theory course for freshmen and sophomores. Topics include prime number theory and cryptographic applications, factorization theory (in integers and Gaussian integers), Pythagorean triples, Fermat descent (for sums of squares and Fermat quartic), Pell's equation, and Diophantine approximation. P/NP or letter grading
Textbook
J. Silverman, A Friendly Introduction to Number Theory (4th edition)
Schedule of Lectures
Lecture  Section  Topics 

1 
2, 3 
Parametrization of Pythagoran numbers; note points where unproved assumptions made. 
2 
4, 5.1 
Statement of Fermat. 
3 
6 
Minimal positive elt. of {ax+by} is gcd(a,b). 
4 
7 
Fundamental Theorem of Arithmetic. 
5 
8 
Congruences. 
6 
9, 10 
Fermat's Little Theorem. 
7 
11 
Chinese Remainder Theorem 
8 
12, 13 
Prime numbers. 
9 
Review 

10 
Midterm #1 

11 
14 
Mersenne Primes. 
12 
16 
Powers mod m and squaring 
13 
17, 18 
Roots mod m. 
14 
19 
Primality testing. 
15 
Powers mod p and primitive roots: show lcm of orders of set of generators = p1; existence of element of order = lcm. 

16 
23 
Squares mod p. 
17 
24 
Square roots and quadratic reciprocity: case of 1. 
18 
25 
Quadratic reciprocity. 
19 
26 
Primes congruent to 1 mod 4 are squares (descent). 
20 
27 
Integers that are sums of two squares. 
21 
Review 

22 
Midterm #2 

23 
28 
Fermat Quartic descent. 
24 
33 
Gaussian integers: basic properties. 
25 
34 
Gaussian integers have unique factorization. 
26 
34 
Application to representation numbers for sums of two squares. 
27 
31 
Diophantine Approximation. 
28 
32 
Pell's equation. 
29 
Review 