MATH 319 Ė 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 ďby Theorem 3Ē.  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. Vassilevís 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

Eulerís Theorem and Wilsonís Theorem, Solving congruences and computing high powers modulo n

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

Eulerís Criterion

 

3/23

4.3

Gaussí Lemma

 

3/25

4.3

Gaussí 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