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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.
Triples via rational parametrization of circle.

2

4, 5.1

Statement of Fermat.
Euclidean algorithm.

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.
Euler's Theorem.

7

11

Chinese Remainder Theorem

8

12, 13

Prime numbers.
Counting primes

9

Review

10

Midterm #1

11

14

Mersenne Primes.
Mersenne Primes and perfect numbers.

12

16

Powers mod m and squaring

13

17, 18

Roots mod m.
RSA

14

19

Primality testing.

15

Powers mod p and primitive roots: show lcm of orders of set of generators = p-1; 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