MAT315H1S: Introduction to Number Theory Winter 2021
I. Instructor and Teaching Assistants
Course Instructor
Name: Stephen Kudla
Email: [email protected]
Office Hours: W 8-9pm
TA’s
Name: Enrique Nunez-Lon-wo
Email: [email protected]
Office Hours: = tutorial hours
Name: Georgios Papas
Email: [email protected]
Office Hours: = tutorial hours
Name: Shuyang Shen
Email: [email protected]
Office Hours: =tutorial hours
Name: Artane Siad
Email: [email protected]
Office Hours: = tutorial hours
II. Course Overview
Course Description
Introduction to Number Theory
Elementary topics in number theory: arithmetic functions; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadratic reciprocity law, representation of numbers as sums of squares.
Prerequisites
(MAT223H1/MATA23H3/MAT223H5/MAT240H1/MAT240H5, MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5,MAT246H1/CSC236H1/CSC240H1)/MAT157Y1/MAT157Y5/MAT247H1/MAT247H5
Course Objectives
This course will provide a basic knowledge of elementary number theory, a subject that is important in many areas of mathematics, computer science, cryptography and security, and elsewhere. We will cover material from Chapters 1—7 of the text by Jones and Jones and will pick up some additional material to be covered in handouts. The main topics are the following:
1. divisors, Euclidean algorithm, gcd(a,b), lcm(a,b).
2. primes, factorization, Prime Number Theorem (statement), Fermat and Mersenne primes,
3. modular arithmetic, congruences, Chinese Remainder Theorem,
4. computing powers and roots mod m, intro to RSA
4. arithmetic mod p for p a prime, Fermat's little Theorem, Wilson's Theorem, primality testing, pseudo-primes and Carmichael numbers
5. (\Z/n\Z)^x, the Euler \phi function, \phi(n).
6. structure of (\Z/p^e\Z)^x, primitive roots
7. quadratic congruences, quadratic reciprocity, Legendre symbol
Additional topics as time permits selected from:
-- the Riemann zeta function, Euler product, Riemann hypothesis,
-- irrational and transcendental numbers, Liouville numbers
-- the abc conjecture.
Textbooks/ Course Reading
Required Textbook:
G. Jones and J. M. Jones, Elementary Number Theory, Springer, ISBN-13:978-3540761976.
(An ebook version of this text is available.)
Additional references:
(These books are not required but are nice sources of additional material and perspectives.)
J. Silverman, A Friendly Introduction to Number Theory, Pearson, 4th ed., 2012, ISBN-13:978-0321816191.
A.Granville, Number Theory Revealed: An Introduction, Amer. Math. Soc., 2019.
K. Rosen, Elementary Number Theory, 6th ed., Addison Wesley, ISBN-13:978-0321500318.
Course Website: The address for Quercus course site is: https://q.utoronto.ca/courses/201925
How this course is organized:
Each week there will be three 1 hour lectures: MWF, 10:10—11:00. [All times are Toronto times.]
Each student should enroll in a tutorial section.
Due to the Covid-19 pandemic, all components of the course will be delivered online synchronously on BB Collaborate. A link will be provided through the Quercus course site.
Lectures will be recorded and posted on the Quercus course site following each class.
Additional material will be posted on the Quercus course site.
Tutorials will start the week of Jan 11th and will be synchronous, so students will be expected to attend the tutorial at the scheduled time for their registered section.
The course was originally scheduled in a combined live classroom/online format. Now that it is running online only, the lecture sections 0101 (previously classroom) and 9101 are now the same.
Similarly, for the tutorials, so that there are now 6 tutorial sections:
W11-12: 0101 = 9101, W11-12: 0102 = 9102, W12-1: 0201 = 9201,
F11-12: 0301 = 9301, F12-1: 0401 = 9401, F1-2: 0501 = 9501.
There will be two Midterm Exams, one during week 5 and one during week 10.
There will be weekly homework. These assignments will be posted after the Monday lecture and will be collected on Crowdmark before the start of lecture the following Monday (i.e., due at 10:00am).
Please note: No late homework will be accepted.
Technical Requirements
In order to participate in this course, students will be required to have:
• Reliable internet access. It is recommended that students have a high speed broadband connection (LAN, Cable, or DSL) with a minimum download speed of 5 Mbps.
• A computer satisfying the minimum technical requirements (https://www.viceprovoststudents.utoronto.ca/covid-19/tech-requirements-online-learning/)
Other recommended items include headphones, microphone, webcam, and a tablet or printer.
If you are facing financial hardship, you are encouraged to contact your college or divisional registrar (https://future.utoronto.ca/current-students/registrars/) to apply for an emergency bursary.
III. Evaluation/ Grading Scheme
Mark Breakdown
Homework Assignments 40%
Term Test 1 15%
Term Test 2 15%
Final Assessment 30%
Homework Assignments
There will be 10 homework assignments, of which the three with lowest grades will be dropped.
(Late submissions will receive a mark of zero.).
The due dates will be at 10am sharp on the Mondays of weeks 2 – 5, weeks 6—10, and week 12.
Midterm Tests
There will be two Midterm tests which will be available for 24 hours to allow a reasonable window for students in all time zones to write.
Midterm Test 1 will be on February 10, 7pm ET to February 11 7pm ET
Midterm Test 2 will be on March 24, 7pm ET to March 25, 7pm ET
Final Assessment
The final assessment will be held during the final assessment period in April 2021 and will be scheduled by the registrar. Information about the format will be provided during the Winter semester.
IV. Course Policies
Policy on Missed Term Work
As flexibility for missed or late homework assignments has been built into the marking scheme, late and missed assignments will not be accepted for any reason.
Please note that Verification of Illness forms (also known as a “doctor’s note”) are temporarily not required. Students who are absent for any reason (e.g., COVID, cold, flu and other illness or injury, family situation) and who require consideration for missed academic work should report their absence through the online absence declaration. The declaration is available on ACORN under the Profile and Settings menu.
If you miss a term test or the final assessment, then you must inform your course Instructor within 72 hours of the test. No exceptions. If your request is approved, you may receive an accommodation in the form of a re-weighting of your assessments.
Email Policy
Should you have a question that is not answered on the course site (please check there first!) please note that all communications with the Course Instructor or TA’s must be sent from your official utoronto email address, with the course number included in the subject line. If these instructions are not followed, your email may not receive a response.
V. Institutional Policies and Support
Academic Integrity
All suspected cases of academic dishonesty will be investigated following procedures outlined in the Code of Behaviour on Academic Matters (https://governingcouncil.utoronto.ca/secretariat/policies/code-behaviour-academic-matters-july-1-2019). If you have questions or concerns about what constitutes appropriate academic behaviour or appropriate research and citation methods, please reach out to your Course Instructor. Note that you are expected to seek out additional information on academic integrity from me or from other institutional resources (for example, the University of Toronto website on Academic Integrity http://academicintegrity.utoronto.ca/).
Copyright
This course, including your participation, will be recorded on video and will be available to students in the course for viewing remotely and after each session.
Course videos and materials belong to your instructor, the University, and/or other sources depending on the specific facts of each situation and are protected by copyright. Do not download, copy, or share any course or student materials or videos without the explicit permission of the instructor.
For questions about the recording and use of videos in which you appear, please contact your instructor.
Accessibility
The University provides academic accommodations for students with disabilities in accordance with the terms of the Ontario Human Rights Code. This occurs through a collaborative process that acknowledges a collective obligation to develop an accessible learning environment that both meets the needs of students and preserves the essential academic requirements of the University’s courses and programs.
Students with diverse learning styles and needs are welcome in this course. If you have a disability that may require accommodations, please feel free to approach your Course Instructor and/or the Accessibility Services office as soon as possible. The sooner you let us know your needs the quicker we can assist you in achieving your learning goals in this course.
Link to Accessibility Services website: https://studentlife.utoronto.ca/department/accessibility-services/
Equity, Diversity and Inclusion
The University of Toronto is committed to equity, human rights and respect for diversity. All members of the learning environment in this course should strive to create an atmosphere of mutual respect where all members of our community can express themselves, engage with each other, and respect one another’s differences. U of T does not condone discrimination or harassment against any persons or communities.
Services and Support
Other Academic and Personal Supports
• Writing Centre https://writing.utoronto.ca/writing-centres/arts-and-science/
• U of T Libraries https://onesearch.library.utoronto.ca/
• Feeling Distressed? https://studentlife.utoronto.ca/task/support-when-you-feel-distressed/
• Academic Success Centre https://studentlife.utoronto.ca/department/academic-success/
• College/Faculty Registrars https://future.utoronto.ca/current-students/registrars/
VI. Schedule of Lectures
Introduction, primes, Fibonacci numbers, induction
gcd’s and lcm’s, Euclidean algorithm
linear Diophantine equations
Prime factorization, p=4k+3, twin primes
prime number theorem, Fermat and Mersenne primes
Primality testing and factorization
Congruences, [a]_n
Linear congruences, Chinese remainder theorem
Chapter 3: 3.4, Chapter 4: 4.1
Fermat’s little theorem, non-linear congruences
Reading Week
Carmichael numbers, Miller’s test
Congruences modulo prime powers, Euler’s phi-function
Units, primitive roots
RSA and cryptography
Chapter 7: 7.2-7.4
Quadratic reciprocity, Legendre symbol
Fermat’s last theorem, abc conjecture
Transcendental numbers, Liouville numbers
review
2021-01-13