Review session TUESDAY DEC 7 2010 at 10am.
If the hallway outside my office (RLM 9.140) is not sufficient we will
go to RLM 10.176.
** Here is the final homework**

Instructor: Dave Rusin (rusin@math.utexas.edu) Office hrs: T, Th 11-2 and by appointment, in RLM 9.140 Text: Number Theory and its Applications (6th Edition) by Kenneth Rosen Lecture: RLM 6.116, T,Th 9:30-11am

Course webpage: http://www.ma.utexas.edu/~rusin/328K/

This is a first course that emphasizes understanding and creating proofs; therefore, it must provide a transition from the problem-solving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to concentrate on developing the students theorem-proving skills.

Homeworks: Homeworks will be assigned on a roughly weekly schedule. Collectively they will count for one-third of the semester grade. Please note that this is not a computational course: there will not usually be "an answer to circle"; your answer to each question will typically be a couple of paragraphs that explain why something is true. So make sure your homework is neat and orderly and is written in complete sentences where appropriate.

Exams: There will be 2 mid-term exams, probably late September and mid-November. Together they will be worth one-third of your semester grade. They will be scheduled several weeks in advance. By universal acclaim, the class expressed a desire to have the exams during the regular meeting time; so be it! The final exam will be cumulative, and also worth one-third of your semester grade.

I will probably not assign plus-minus grades (A-, B+, etc.) since the class seemed to prefer that I not use them. If you would like to express an opinion about this issue, please do so before the first exam.

Students with disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.

Drop dates: The last day to drop the course without possible academic penalty is Sept 22, 2010. For more information about deadlines for adding and dropping the course under different circumstances, please consult the Registrar's web page, http://registrar.utexas.edu/calendars/10-11/

Here is a tentative outline of what we will cover. This is sure to change, for at least two reasons: (a) we have to allow some time for exams! (b) The material of the last few classes is really up to you: which of the topics in the last half of the book seems most interesting to you?

My tentative plan:

- Thursday, August 26: Introduction to number theory, divisiblity, and proofs by definition.
- Tuesday, August 31: Divisiblity, congruences, and modular arithmetic.
- Thursday, September 2: Congruences, modular arithmetic, and applications.
- Tuesday, September 7: Greatest common divisors.
- Thursday, September 9: Proofs by induction and greatest common divisors.
- Tuesday, September 14: Greatest common divisors, linear Diophantine equations, invertible elements, and Chinese remainder theorem.
- Thursday, September 16: Fundamental theorem of arithmetic.
- Tuesday, September 21: Fermat's little theorem, Wilson's theorem, and applications.
- Thursday, September 23: Euler's phi function and Euler's theorem.
- Tuesday, September 28: Euler's phi function and other multiplicative functions.
- Thursday, September 30: Moebius inversion and Legendre symbols.
- Tuesday, October 5: Legendre symbols, Euler's criterion, Gauss' lemma, and special cases of the Legendre symbol
- Thursday, October 7: Midterm
- Tuesday, October 12: Special cases of the Legendre symbol and quadratic reciprocity
- Thursday, October 14: Proof of quadratic reciprocity and the Jacobi symbol
- Tuesday, October 19: Quadratic reciprocity for the Jacobi symbol and primitive roots
- Thursday, October 21: Primitive roots and Lagrange's theorem
- Tuesday, October 26: Existence of primitive roots
- Thursday, October 28: Nonexistence of primitive roots and index arithmetic
- Tuesday, November 2: Power residues, the proof of Chinese remainder theorem, and Hensel's lemma
- Thursday, November 4: The proof of Hensel's lemma and Pythagorean triples
- Tuesday, November 9: Pythagorean triples and Fermat's last theorem
- Thursday, November 11: Cryptography
- Tuesday, November 16: The prime number theorem
- Thursday, November 18: Gaussian integers: definition, associates and units, norm, divisiblity, and primes
- Tuesday, November 23: Gaussian integers: division algorithm and greatest common divisors
- Thursday, November 25: No classs -- Thanksgiving holiday
- Tuesday, November 30: Gaussian integers: unique factorization, sums of squares, and classification of primes.
- Thursday, December 2: Sums of squares, Gaussian primes, and prime number theorems for Gaussian integers