MATH 115A (Number Theory) - Fall 2010, UC Davis

Lectures: MWF 1:10-2:00pm, 158 Olson (Schilling)
Discussion Sessions: MAT 115A-A01, CRN 68686, T 3:10-4:00P in Haring 1204 (Fineman)
MAT 115A-A01, CRN 83500, T 3:10-4:00P in OLSON 207 (Li)
Instructor: Anne Schilling, MSB 3222, phone: 554-2326,
Office hours: Mondays 3-4pm, Wednesdays 2-3pm
T.A.: Ben Fineman, MSB 3123
Office hours: Mondays 10-11am
Lingyun Li, MSB 3139
Office hours: Tuesdays 4:10-5:10pm
Text: I will mostly follow K.H. Rosen, Elementary Number Theory, Addison-Wesley, ISBN 0-321-23707-2, but I will not require students to purchase the book!
Another very good book, which is available free of charge, is by William Stein Elementary Number Theory: Primes, Congruences, and Secrets. We will use this book in particular for Sage examples.
Pre-requisite: MAT 21B
Problem Sets: There will be weekly homework assignments due on Wednesdays at the beginning of class.
You are encouraged to discuss the homework problems with other students. However, the homeworks that you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources. If you need help with the problems, come to the discussion session and office hours! The best way to learn mathematics is by working with it yourself. No late homeworks will be accepted. A random selection of homework problems will be graded.
Computing: During class, I will illustrate some results using the open source computer algebra system Sage. When you follow the link, you can try it out yourself using Sage Online Notebook. Or you can sign up for a Class Account with the math department. Log into and type the command `sage` to launch a Sage session in the terminal.
Exams: There will be one Midterm on October 27 in class and one Take-home Midterm in the week of November 15. The Final exam will be Wednesday, December 8, from 8:00-10:00am, in 179 Chemistry (not the usual classroom!).
There will be no make-up exams!
Grading: The final grade will be based on: Problem sets 20%, Midterms 20% each, Final 40%.
Grades will be recorded on SmartSite.
Bed Time Reading: If you would like some bedtime reading related to this class, I can recommend two really good books by Simon Singh: "Fermat's Last Theorem" and "The Code Book".

Course description

This course is the first part of a two-quarter introduction to Number Theory. Number theory is the study of properties of numbers in particular the integers and rational numbers. Questions in elementary number theory include divisibility properties of integers (e.g. the Euclidean algorithm), properties of primes (e.g. there are infinitely many), congruences, quadratic reciprocity and integer solutions to basic equations (e.g. Diophantine equations). Even though number theory is one of the oldest disciplines in mathematics, it has recently contributed to many practical problems such as coding theory, cryptography, hashing functions or other tools in modern information technology. These applications will also be part of this class! The class is primarily based on Chapters 1-8 of Rosen's book, but we will also refer to Chapters 1-3 of Stein's book.

1. Prime factorization
prime numbers, Euclidean algorithm, the fundamental theorem of arithmetic, factorization methods, linear diophantine equations

2. Congruences
linear congruences, Chinese remainder theorem, Wilson's, Fermat's and Euler's theorem, Euler's Phi-function

3. Applications to Congruences (time permitting)
divisibility tests, hashing functions, public-key cryptography

Sage Examples

Distribution of Primes
Carmichael Numbers
RSA Algorithm
Discrete Logarithm

Problem sets

Homework 0: (voluntary) send me an email about yourself, your goal and expectations for the class or anything else you would like to share!
Homework 1: due October 6, 2010 in class: pdf
Solution: pdf
Homework 2: due October 13, 2010 in class: pdf
Solution: pdf
Homework 3: due October 20, 2010 in class: pdf
Solution: pdf
Homework 4: due October 27, 2010 in class: pdf
Solution: pdf
Midterm: October 27, 2010
Solution: pdf
Homework 5: due November 3, 2010 in class: pdf
Solution: pdf
Homework 6: due November 10, 2010 in class: pdf
Solution: pdf
Homework 7: due November 17, 2010 in class: pdf
Solution: pdf
Take Home: due November 19, 2010 in class: pdf
Homework 8: due December 1, 2010 in class: pdf
Solution: pdf
Review and Practice Problems: discussed in discussion session November 30: pdf