# Syllabus,  schedule and homework.

Slides and other materials can be found here.

This schedule will be regularly updated. It is your responsibility to check it accordingly.

Homework is due every Thursday. Underlined problems in should be handed in.

 When Topics # Homework, Exams, Remarks. 1/24 Administrative stuff 1.1 Introduction to Number Theory, 1.2 Divisibility: greatest common divisor, Euclidean Algorithm 1.3 Primes 0 Fill this form. Corrections to the textbook are here and here. 1/31 Distribution of primes. 1.4 The binomial theorem 1 Section 1.2. Problems 15 and 32; Section 1.3. Problems 11 and 31. and some problems of your choice to practice. 2/7 2.1 Congruences 2.2 Solutions of congruences 2 Section 1.3: 22, 24, 25, 27, 30 Section 1.4: 1,4, 5, 8 I).Show that there are infinitely many primes of the form 6n+5. b. Using the idea of the proof of the "Infinitely many primes theorem", try  to prove that  there are infinitely many primes of the form 5n+4. Something goes wrong... what is it? II) Explain why the statement "One sixth of the numbers have reminder 3 when divided by 6 makes sense". III) Explain why the statement "most numbers are not perfect squares" makes sense. Find a function (defined in terms of  a simple formula) to approximate the counting funcion C(x) (see below) when x is large. In problems II) and III) use a counting function of the form C(x)=#{n is a positive number, n ≤ x, n satisfies a certain property } (you need to determine which property). Announcements. 1. The university have cancelled classes tomorrow. Homework set 2 (the one originally due Feb  9) is now due on Thursday Feb 16 (together with Homework set 3. 2. I will arrive late to my office hours on Wednesday Feb 15th (I guess sometime before 11am). I will be available later. Send me an email if you need help. 2/14 2.1 and 2.2 Review and complete. 2.3 The Chinese Remainder Theorem 3 Section 2.1: 2, 4, 5, 7, 9, 13, 19, 25, 40 Section 2.2: 5, 6, 7, 9 , Note: The problems marked with an H have a hint at the end of the book. Try first to solve the problems  without reading the hint. 2/21 2.4 Techniques of numerical calculation (excluding Pollard rho method) 4 Section 2.3: 1, 2, 9, 10, 16. 2/28 2.5 Public key cryptography 5 Section 2.4: 2, 4, 9 Section 2.5: 1 (Solve this problem using the Euclidean algorithm), 2, 3, 4, 5 Write down a list of the main results we discussed in class. in your own words. in one page. Write down the main ideas of the proof of some (or all) of the statements. 3/7 Midterm Guest Lecture: Continued fractions 6 Tuesday 3/7 Midterm 1 In class (Note the change of date). Midterm topics EVERYTHING until Public Key Cryptography (including Public Key Cryptography) No homework this week, but here are a list of problems you should know how to solve. Section 2.8: 1, 2, 3, 4, 5, 8,  12, 3/14 Spring break 3/21 2.8 Primitive roots and power residues 7 Section 2.8: 1, 2, 3, 13, 14,15, 18, 20. 3/28 7.1 The Euclidean algorithm 7.2 Uniqueness 7-8 Here are "half"of the problems to submit Section 7.1: 1,2,3,4,5 4/4 7.3 Infinite continued fractions Review 9 Here are the problems to submit.  Sample problems for the midterm. 4/11 Midterm 2 7.3 Infinite continued fractions 10 Tuesday 4/11 Midterm 2 in class. No homework this week. 4/18 3.1 Quadratic residues 3.2 Quadratic reciprocity 11 Problems for this week are here. Hand in problems 3, 4, 5, and 6. Try to submit one or more of the bonus problems. (You are allowed to work in teams for the bonus problems). 4/25 3.2 Quadratic Reciprocity 4.1 Some functions in number theory 4.2 Arithmetic functions. Mersenne Primes 4.3 The Mobius inversion formula 12 Problems for this week are here. Hand in problems 5, 6, 7 and 8. Goody Bag Pascaline You can see this movie in a smart phone with the Puffin Web Browser. The site has an enormous amount of wonderful math related apps (in French). An interesting movie about Mersenne. 5/2 The Riemann Hypothesis and the distribution of primes (an interesting article can be found here) Visualizing the Riemann zeta function and analytic continuation 13 The music of primes movie. Section 4.2:  2, 3, 5, 9, 13, 14, 17, 20 Section 4.3: 1, 2, 3, 5, 13, 18, 19,    Final exam practice. Final Exam: Monday, May 15, 11:15am-1:45pm, in our usual classroom, N4000 at the library.