THEORY OF NUMBERS MATH 115B, Course Information

Meetings: MWF 11:00pm-11:50 AM, Room 233 Wellman Hall.

Instructor: Jesús A. De Loera. email: deloera@math.ucdavis.edu http://www.math.ucdavis.edu/d̃eloera/TEACHING/MATH115B/courseinfo/115B.html Phone: (530)-754 70 29

Office hours: Wednesday 12:10pm-2:00pm or by appointment. My office is 580 Kerr Hall. The TA for this class is Mr. Matthew Hubbard, his office hours will be announced soon. I will be glad to help you with any questions, concerns or problems you may have!

Text: I will not require any particular textbook and I hope my lecture notes should help you with most of the material. In the first five weeks I will follow many books (some not even in English) so my lecture notes will be the main source. In the second part I will mostly follow the last 5 chapters of “Elementary Number Theory and its applications” by Kenneth H. Rosen Addison Wesley, but I will not require that students purchase the textbook. Other excellent references are:

Hardy, G. H.; Wright, E. M.: ”An introduction to the theory of numbers”, The Clarendon Press, Oxford University Press. Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L.: ”An introduction to the theory of numbers”, John Wiley & Sons, Inc. Description: Number Theory regards the study of properties of numbers, more particular integer or rational numbers. In this course I will try to concentrate on Number Theory relevant in coding theory, cryptography, and other parts modern information technology.

Topics to be covered in 115B (Winter 2003):

1) Lattices points on lines, circles and polygons.

2) Pick’s theorem and its generalizations.

3) Lattices

4) Minkowski’s geometry of numbers

5) Applications: Diophantine approximation, Short vectors.

6) Primitive roots. ElGamal Cryptosystem.

7) Quadratic residues, Jacobi symbol, sums of squares

8) Continued fractions and Pell’s equation.

9) More on Non-linear Diophantine Equations. Grading policy: There are 100 points possible in the course.

• There will be 2 midterms, first on February 3rd 2003 and the second March 5th 2003. ach counting 35 points. I will drop the lowest of the two for the calculation of the grade. Use of books, notes, and calculators will not be allowed on any exam. The final exam is comprehensive and counts 40 points.

• Homework will be assigned and collected almost every week, on Wednesdays, for a total of 7 homeworks (about 5 problems in each homework). Each homework is worth 5 points. Two of the problems will be chosen for grading but you will receive one point if you wrote solutions for all problems assigned. The remaining 4 points will be awarded in the 2 problems being graded. I will drop the two lowest scores from the calculation of the final grade, this is for a maximum of 25 points.

• I will assign grades based on an statistical information of the points obtained by all students (I compute the mean, standard deviation, etc. and set letter grades according with those numbers), but typically 60 points or higher are required to pass the class.

• IMPORTANT I will handle all grades via the myucdavis grade system. This means that if you are registered students at UC Davis you can access grade information for this class via the internet (check https://my.ucdavis.edu/ for details). This is a secure and private web page assigned to each student. You can see your standing in the class, important statistics on exams, and your final grade there! I will not disclose your grade in any other form.

Important rules:

• New homework exercises will be posted on my web page (see the end of the page). I may write a few new problems after each class, so please check the web page often. Please remember there are NO make-up exams or quizzes, instead I am dropping the lowest score as a compensation of possible problems or emergencies. Graded homeworks will be distributed in class. Please check the course web page often!

Prerequisites and Expectations: MAT 115A, Mat 108 or equivalent are a pre-requisite. If in doubt, please ask me. You are expected to work intensively outside the classroom solving exercises, reading the book, thinking about the theorems, etc. I estimate a minimum of 2 hours work at home per lecture. The most important thing is what YOU learn. Mathematics is fun and pretty, try to get the material in your soul! Rote memorization of facts is useless and you are expected to think about the material everyday. It is easy to fall behind, please be careful! I am here to help you and I hope you enjoy the course.

HOMEWORK: Check the web page!