THEORY OF NUMBERS MATH 115A, course information

Meetings: MWF 11:00pm-11:50 AM, Room 212 VEIMYR

Instructor: Jesús A. De Loera. email: deloera@math.ucdavis.edu http://www.math.ucdavis.edu/d̃eloera/TEACHING/MATH115A/courseinfo/115A.html Phone: (530)-754 70 29

Office hours: Monday and Friday 12:10pm-2:00pm or by appointment. My office is 580 Kerr Hall. The TA for this class is Ms. Lipika Deka, here office hours are Tuesday: 3.30-4.30pm Thursday: 1.30-2.30pm. I will be glad to help you with any questions, concerns or problems you may have!

Text: I will mostly follow “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: This is the first part of a year-long undergraduate-level course in Number Theory. Number Theory regards the study of properties of numbers, more particular integer or rational numbers. Elementary number theory involves divisibility among integers (e.g. the Euclidean algorithm), elementary properties of primes (e.g. there are infinitely many!), congruences, quadratic reciprocity and the solution by integers of basic equations. It also touches beautiful and famous number sequences such as the Fibonacci numbers or the Pythagorean triples. This course would cover most of these topics. Of course, Number Theory, being one of the oldest areas of mathematics, has already borrowed weapons from geometry, analysis, and algebra. When possible, I will try to at least glance at such ideas. Besides introducing you to such classical material, I will try to make you aware of the applied importance of the field, since Number Theory is relevant in coding theory, cryptography, hashing functions, and other tools in modern information technology.

Topics to be covered in 115A (Fall 2002):

1) Prime Numbers and the Euclidean Algorithm. 2) The fundamental Theorem of Arithmetic. 3) Factorization methods. 4) Congruence and the Chinese Remainder Theorem. 5) Special Congruences.

Winter and Spring topics will include: Euler-Phi function, Mobius inversion, Public Key encryption, Primitive roots, quadratic residues, continued fractions, Lattices, Minkowski Geometry of Numbers. Grading policy: There are 100 points possible in the course.

• There will be 2 midterms, first on October 25th 2002, the second on November 25th 2002 each 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 10 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 the boxes in Wellman hall or in recitation. Please check the course web page often!

Prerequisites and Expectations: MAT 21D, 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 3 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: It is already posted in the web site!