Number Theory MS-E1110 (5 cr) Course Presentation Lecturer: Camilla Hollanti TA: Taoufiq Damir
Contact emails: {mohamed.damir, camilla.hollanti}@aalto.fi Taoufiq’s room: Y249a (reception hours Tuesdays 14-16) Camilla’s room: Y239 (appointment by email) For more information on the ANTA research group, see http://math.aalto.fi/en/research/discrete/anta/ Number Theory MS-E1110 (5 cr) Course Presentation 2/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
Tentative Program The first 4 weeks (the usual): L1–L2 Primes : Fundamental Theorem of Arithmetic. Euclidean division algorithm and GCD. ([1, Ch. 1]) L2–L4 Modular Arithmetic : Chinese Reminder Theorem, Euler’s ϕ function, Fermat’s Little Theorem, Euler’s Theorem. ([1, Ch. 2]) L4–L8 Quadratic Reciprocity Law : statement and proof. ([1, Ch. 4]) L8 Application : RSA public key crypto system (time permitting). ([1, Ch. 3]) The last 2 weeks will consist of project work and presentations with help from the TA. No lectures/exercises on the weeks 5–6, and on the 6th week these hours are used for the presentations. [1] W. Stein, Elementary Number Theory: Primes, Congruences and Secrets (pdf on MyCourses) Number Theory MS-E1110 (5 cr) Course Presentation 3/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
Grading System 3 homework assignments: 20+25+25=70 % Final Project: 30 % Note that both parts are mandatory! You cannot pass the course with only homework/project. No exam. Some Advise Attendance is recommended. Number Theory is a vast subject and this course gives a good basis for further studies, e.g. related to more advanced number theory or cryptography. It is worthwhile to take advantage of the experience of the teachers. We will closely follow the reference book [1] so read the corresponding chapters. Number Theory MS-E1110 (5 cr) Course Presentation 4/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
About the Final Project Around the end of the second week we will propose a list of possible projects among which you can choose. One topic can be chosen by several students. The project work can be realized individually or in pairs. The depth and length should reflect this. The project consists of self-study and a powerpoint presentation to the class. Genuine effort will be rewarded. More details to follow. Number Theory MS-E1110 (5 cr) Course Presentation 5/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
Why Project? This project is meant for practicing your research skills. In real (working) life, you are not always presented with well-defined problems, and especially not with well-defined solutions. Finding material independently and critically rather than being provided with explicit references is also a good skill, and not only in the academia. If you feel lost at points and not sure what to do, don’t worry, that’s how research is (and you can always turn to your TA for help). Number Theory MS-E1110 (5 cr) Course Presentation 6/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
Further Recreational/Popular Readings Marcus du Satoy, The Music of the Primes Simon Singh, Fermat’s Last Theorem Mario Livio, The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry See also the slide set from the 2016 course edition (some nice NT history etc.) on MyCourses. Some Classical References Hardy–Wright, An Introduction to the Theory of Numbers Apostol, Introduction to Analytic Number Theory Davenport, Higher Arithmetic: an Introduction to the Theory of Numbers Number Theory MS-E1110 (5 cr) Course Presentation 7/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
If you fall in LOVE with Number Theory (1) Course on Galois Theory (5 cr, Period IV/2020) (2) Course on Algebraic Number Theory (5 cr, Period V/2020(?)) (3) BSc/MSc thesis with a member of the ANTA group (4) ANTA Seminar (2 cr, continuous): http://math.aalto.fi/en/research/discrete/anta/ seminar.php Number Theory MS-E1110 (5 cr) Course Presentation 8/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
A Problem from the Mathematical Olympiad Prove that for every prime p � = 2 , 5 there exists an integer n ∈ Z of the form n = 111 . . . 111 such that p | n . Solution: At some point of the course ... Number Theory MS-E1110 (5 cr) Course Presentation 9/9 Lecturer: Camilla Hollanti TA: Taoufiq Damir 11.9.2019
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