MATH115A-f07

Number Theory

MATH 115A, course information
http://www.math.ucdavis.edu/~santos/MATH115A

Announcement: Solutions to the mid-term test here.

Homework 5 is posted, and due November 9. Click here to go to the homework page.

Class Meetings: MWF 12:10-13:00, Room 1062 BAINER
Class Discussion: Thu 12:10-13:00, Room 1062 BAINER

Instructor: Francisco Santos.
email: s a n t o s @ m a t h . u c d a v i s . e d u
Phone: (530)-752 31 85
Office hours: MWF 10:00-12:00 or by appointment, in office 3136 in the Math Science Building.
TA for this course: David Haws. His office hours in 3127 MSB are Thursdays, 1 to 2 pm.

Text: The book I will follow is

``Elementary Number Theory and its applications'' by Kenneth H. Rosen, (fifth edition), Addison Wesley, 2005.

The book has its own website where you can find, among other things, a student manual with solutions to selected exercises. I recommend, but do not require, that students purchase the textbook. (If you do, it will serve you for 115B as well). There are many other good books out there that can help you follow the course. For example:

Burton, D. M.: "Elementary Number Theory", McGraw-Hill.
Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L.: "An introduction to the theory of numbers", John Wiley & Sons, Inc.
Hardy, G. H.; Wright, E. M.: "An introduction to the theory of numbers", The Clarendon Press, Oxford University Press.

Description: This is the first part of an 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 we will try to at least glance at such ideas. Besides introducing you to such classical and old material we 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)

Essentially, the course covers Chapters 1, 3, 4 and 6 of Rosen's book, together with parts of 7 and 8. A list of topics is:

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.

You can access the timelined syllabus suggested by the Math department, which I will try to follow.
If you are interested, 115B and 115C topics, taught in Winter and Spring, include: Euler-Phi function, Mobius inversion, Public Key encryption, Primitive roots, quadratic residues, continued fractions, Lattices, Minkowski Geometry of Numbers.

Grading policy: There are 110 points possible in the course, distributed as follows:

25 points assigned for homework. There will be some 7 or 8 homework assignments during the course, and each will be graded to a maximum of 5 points. Your homework grade will be the sum of the grades in the best five homeworks you submit.
36 points assigned to the mid-term exam. The mid-term will take place on Friday November 2nd, during the class hour. It will include whatever material we have covered in class up to that point. This will probably mean Chapters 1 to 3 and perhaps part of 4.
49 points for the Final exam. The final exam is scheduled for Friday December 14, at 8:00 am. It will include the whole course.

Homework rules

Homework is due IN CLASS by the date it is assigned. Graded homeworks will be distributed in the class as well.
Each homework will consist of 8-10 problems. Three or four of them will be chosen to be graded, to a maximum of 4 points. One extra point will be given for turning all of the solutions, so that each homework is worth 5 points. Your five best homeworks will give your homework grade, with a maximum of 25 points.

Exam rules

Use of books, notes, calculators and cell phones will not be allowed on any exam.
All students must bring their student ID to the exams, and place it on the table.
FINAL EXAM DATE: December 14, 8:00 am. Please mark your calendars! and don't make other plans.

Grades

I will assign grades based on a statistical information of the points obtained by all students (I compute the mean, standard deviation, etc. and set letter grades according with those numbers). I expect that at least 65 points (remember, out of 110) will be necessary to pass this course.

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 in a secure and private web page assigned for 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:

YOU are responsible for reading the textbook regularly, moving along on the text as we advance through the different sections. A new section is started almost every class.
New homework exercises will be posted on the web page. I may write a few new problems after each class, so please check the web page often. Please check often the course web page often!

Prerequisites and Expectations: MAT 21B, 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! It is easy to fall behind, please be careful!

I am here to help you, I will be very happy to talk to you about any question or idea you had and I hope you will enjoy the course!