ÕækQË@áÔgQË@é<Ë@Õæ. Introduction to Number Theory Syllabus Spring 2010 This course is going to run on-line. You can attend the course in two dierent ways; one way is by going to the designated room (will be determined in your timetable) at the University of Nizwa, and the second way is by using your own laptop or PC and going on-line to the site of the course which provided by "Wiziq" and login using your username and password which you will get when you get an invitation by e-mail. The lectures will be called Introduction to Number Theory Lecture 1, Lecture 2, ... etc. For the rst time you need to register your information such as name, email, .... If you have any problem, please, contact me or Miss Omaima. You can get the information for the course from two sources; One source is from me directly by using email or the second is Miss Omaima. The instruction on how, when and where to attend the class will be handed to the students by Miss Omaima. She will have all the information needed for the course, such as HW, practice sheets, midterms, and nal. She will collect the HW, practice sheets, midterms, and nal from the students, however, some students may choose to email some of these material for me directly. For those who choose to send HW, or any other material to me directly, you can either write the document by hand, scan and then send it as a pdf le or send an electronic version if you have it written using microsoftword or any other word processor. It is recommended that you attend the lectures on time. Lectures will be presented live with audio- video support. I will give the chance for the students to ask questions and ask them to answer some questions that I pose. Lectures will be recorded in order that you may replay them at any time later, so it is recommended that you do not busy yourself with taking notes. If you have a question you may ask and you will be given the mic and allowed to talk. Department Department of Mathematical and Physical Sciences, College of Arts and Sciences, University of Nizwa Course Name & Code MATH340 Introduction to Number Theory Prerequisite MATH221 Foundation of Mathematics. Any student who did not pass Math221 will do very poor in this course, and most likely will fail the course. Students must have a solid background on Methods of proofs, logic, functions, and other elementary stu. 1Instructors Professor Mohammed Elatrash; Main lecturer Ass. Professor Mahjoob Yahya ; He will replace me if any problem arises and I can not continue on-line. Miss Omaiyma Al-Balushi; She will help students how to use the internet to be able to attend, ask a question, send a HW, or any other question until students are familiar to the system and can do it independently. Also, she will have all course material Contacts melatrash@math.ualberta.ca, & m.alatrash@unizwa.edu.om Course Material Lecture Notes will be available. Reference 1. Jones, G.A., Jones, J. M., Elementary Number Theory, Springer SUMS, 1998. 2. Mollin, R. A., Fundamental Number Theory with Applications, CRC Press, 1998. 3. Niven, I., Zuckerman, H., Montgomery H., An Introduction to the Theory of Numbers, Wiley. 4. Rosen, K. H., Elementary Number Theory and its applications, Pearson International, 5th ed, 2005. 5. Gilles Brassard, Modern Cryptology: A Tutorial, Lecture Notes in Computer Science, vol. 325, Springer-Verlag, 1988. Oce Hours 12:00-1:00 Saturday , & 12:00-1:00 Wednesday Oce hours will be on demand; means if you need some assistance you have to e-mail me before you go on-line in order for me to launch a session for you Lectures 8:00-9:00 AM Saturday & Monday. Room: to be determined later in the timetable. I will use lecture time to introduce denitions, new concepts, theorems, propositions, examples and proofs. Tutorial 8:00-10:00 AM Wednesday. I will use tutorial time to solve exercises, Students are expected to participate in problem solving. I will either ask students to solve a problem on the white board or solve it on a sheet of paper. This is not a test, so students can work together, to discuss, solve, and come up with new ideas. Usually, a practice sheet will be provided to students on Mondays in order to work on on Wednesdays. So, have some time to take a look at these problems before you come to class to save class time for the sake of solving as many problems as we can. After all, students 2are responsible to solve these problems. Usually my exams will be similar to those problems in the practice sheets in addition to some of the theoretic material presented in the lectures. Your participation in the tutorial will determine 20 % of the total grade of the course. Course Assessment20% Participation 40% 2 Mid-Term Exams , 40% Final Exam , Weekly Outline Week 1 1 Basic Axioms for Z 2 Proof by Induction Week 2 3 Elementary Divisibility Properties 4 The Floor and Ceiling of a Real Number Week 3 5 The Division Algorithm 6 Greatest Common Divisor Week 4 7 The Euclidean Algorithm 8 Bezout's Lemma Week 5 9 Blankinship's Method 10 Prime Numbers Week 6 11 Unique Factorization 12 Fermat Primes and Mersenne Primes Week 7 13 The Functions and 14 Perfect Numbers and Mersenne Primes Week 8 15 Congruences First midterm Week 9 16 Divisibility Tests for 2, 3, 5, 9, 11 17 Divisibility Tests for 7 and 13 Week 10 18 More Properties of Congruences Week 11 19 Residue Classes 20 Zm and Complete Residue Systems Week 12 21 Addition and Multiplication in Zm 22 The Groups Um Week 13 23 Two Theorems of Euler and Fermat Second Midterm Week 14 24 Probabilistic Primality Tests 25 The Base b Representation of n Week 15 26 Computation of aN mod m Week 16 27 The RSA Scheme Final Exam 3Objectives [1] Students Learn and master the abstract denitions of in the subject. [2] Students Will be able to provide examples. [3] Students Will be able to apply the concepts in other elds. [4] Students Will understand some concepts with abstraction [5] Students will understand the basic ideas and some applications of Integers. [6] Students will Master Modular Arithmetic [7] Students will be able to reason mathematically, to write simple proofs, [8] Students will learn the Fundamental Theorem of Arithmetic . [9] Students will be able to state some special theorems such as ; Fermat's Theorem, Euler's Theorem [10] Students will have a chance to apply the The theory into practice [11] Students will be able to derive and apply division criterion for integers written base 10, 5, and others. [12] Students will be able to calculate many number-theoretic functions such as, greatest common divisors, least common multiples, Euler's function For Ocial Copy Only Signature Course Coordinator Prof. Mohammed Elatrash Date: Head of Section Prof. Ahmed Al-Karamani Date: Head of Department Prof. Ahmed Al-Karamani Date: Dean Prof. Mohammed Ismail Date: 4