MATH
319 – Number Theory
Spring 2011
Professor:
Dr. Janet Vassilev
Office: SMLC 324
Office
Hours: MWF 2-3 pm and by appointment.
Telephone: 
(505)
277-2214
email: jvassil@math.unm.edu
webpage: http://www.math.unm.edu/~jvassil
Text
:
Elementary Number Theory: Primes, Congruences
and Secrets by William Stein.
Course
Meetings: The course lectures will be held in SMLC 356 on Mondays,
Wednesdays and Fridays at 11-11:50 am.
Topics: Divisibility, congruences,
primitive roots, quadratic residues, diophantine
equations, continued fractions, partitions, number theoretic functions.
Homework
(200 points): Homework will be assigned weekly on Fridays and will be
collected the following Friday at the beginning of class. Homework will
not be graded unless it is written in order and labeled
appropriately. The definitions and theorems in the text and
given in class are your tools for the homework proofs. If the theorem has a name, use it. Otherwise, I would prefer that you fully
describe the theorem in words that you plan to use, than state “by Theorem
3”. Each week around 4 of the assigned problems will be graded. The
weekly assignments will each be worth 20 points. I will drop your lowest two homework scores
and the remaining homework will be averaged to get a score out of 200.
Exams (400
points): I will give two midterms (100 points) and a final (200
points). There are no make up exams. If a test is missed,
notify me as soon as possible on the day of the exam.
For the midterms only, if you have a legitimate and documented
excuse, your grade will be recalculated without that test. The midterms
are tentatively scheduled for Friday, February 25 and Friday, April 8.
The Final is on Wednesday, May 11, from 12:30 pm-2:30 pm.
Grades: General guidelines
for letter grades (subject to change; but they won't get any more strict):
90-100% - A; 80-89% - B; 70-79% - C; 60-69% - D; below 60% - F. In
assigning Final Grades for the course, I will compare your grade on all course
work (including the Final) and your grade on the Final Exam. You will
receive the better of the two grades.
Tentative
Schedule (for Dr. Vassilev’s Number Theory):
Date
|
Section
|
Topic
|
Homework
|
1/19
|
|
Introduction,
Induction
|
|
1/21
|
1.1.1
|
Prime Numbers
|
HW 1
|
1/24
|
1.1.2
|
Division Algorithm
|
|
1/26
|
1.1.2
|
Euclidean
Algorithm
|
|
1/28
|
1.1.3-1.2.3
|
Fundamental
Theorem of Arithmetic, cardinality of primes, prime sieve and Mersenne primes
|
HW2
|
1/31
|
1.2.4-1.2.5
|
Primes of the
form ax+b and the Prime Number Theorem
|
|
2/2
|
|
Classes
canceled due to closure
|
|
2/4
|
|
Classes
canceled due to closure
|
|
2/7
|
2.1
|
Congruence
modulo n,
|
|
2/9
|
2.1
|
Congruence
modulo n and solving equations modulo n
|
|
2/11
|
2.1, 2.3
|
Euler’s Theorem
and
|
HW3
|
2/14
|
2.2
|
Chinese
Remainder Theorem
|
|
2/16
|
2.2
|
Chinese Remainder
Theorem
|
|
2/18
|
2.5
|
Order and
Primitive roots
|
HW4 due Monday 2/28
|
2/21
|
2.5
|
Primitive roots
|
|
2/23
|
|
Review
|
|
2/25
|
|
Midterm 1
|
|
2/28
|
2.5,3.2
|
Primitive roots
and Diffie Hellman
|
|
3/2
|
3.2
|
Diffie Hellman and RSA
|
|
3/4
|
3.3
|
RSA and ciphers
|
HW5 p 46-47
2.17, 2.25, 2.27a, 2.28, 2.30, 2.33
P 67 3.4 and
3.5
|
3/7
|
3.4
|
Attacking RSA
|
|
3/9
|
3.4
|
Attacking RSA
|
|
3/11
|
4.1
|
Quadratic
reciprocity
|
HW6
|
3/21
|
4.2
|
Euler’s
Criterion
|
|
3/23
|
4.3
|
Gauss’ Lemma
|
|
3/25
|
4.3
|
Gauss’ Lemma
|
HW7
|
3/28
|
4.4
|
Gauss Sums
|
|
3/30
|
4.4
|
Gauss Sums
|
|
4/1
|
4.5
|
Square roots
|
HW8
|
4/4
|
5.1
|
Continued
Fractions
|
|
4/6
|
|
Review
|
|
4/8
|
|
Midterm 2
|
|
4/11
|
5.2
|
Continued Fractions
|
|
4/13
|
5.3
|
Convergence of
infinite continued fractions
|
|
4/15
|
5.5
|
Quadratic
irrationals
|
HW9
|
4/18
|
5.7
|
Sums of Two
Squares
|
|
4/20
|
5.7
|
Sum of Two
Squares
|
|
4/22
|
6.1-6.2
|
Elliptic curves
|
HW10
|
4/25
|
6.2
|
Elliptic curves
|
|
4/27
|
6.3
|
Integer
factorizations using elliptic curves
|
|
4/29
|
6.3
|
Integer
factorizations using elliptic curves
|
HW11
|
5/2
|
|
Review
|
|
5/4
|
|
Review
|
|
5/6
|
|
Review
|
|
5/11
|
|
Final exam
|
|