MAT315H1S: Introduc3on to Number Theory Fall 2024

Java Python MAT315H1S: Introduc3on to Number Theory

Fall 2024

I.                    Course Overview

Course DescripFon:

IntroducFon to Number Theory

Elementary topics in number theory: arithmeIc funcIons; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadraIc reciprocity law, representaIon of numbers as sums of squares.

Prerequisites

(MAT223H1/MATA23H3/MAT223H5/MAT240H1/MAT240H5,MAT235Y1/MAT235Y5/ (MAT232H5,MAT236H5)/ (MATB41H3,MATB42H3)/MAT237Y1/

(MATB41H3,MATB42H3,MATB43H3)/MAT237Y5,MAT246H1/CSC236H1/CSC240H1)/MAT157Y1 /MAT157Y5/ (MAT157H5,MAT159H5)/MAT247H1/MAT247H5

Course ObjecFves

This course will provide a basic knowledge of elementary number theory, a subject that is important in many areas of mathemaIcs, 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 addiIonal material to be covered in handouts.

The main topics are the following:

1. divisors, Euclidean algorithm, gcd(a,b), lcm(a,b).

2. primes, factorizaIon, Prime Number Theorem (statement), Fermat and Mersenne primes,

3. modular arithmeIc, congruences, Chinese Remainder Theorem,

4. compuIng powers and roots mod m, intro to RSA

4. arithmeIc mod p for p a prime, Fermat's lidle Theorem, Wilson's Theorem, primality tesIng, pseudoprimes and Carmichael numbers

5. (\Z/n\Z)^x, the Euler \phi funcIon, \phi(n).

6. structure of (\Z/p^e\Z)^x, primiIve roots

7. quadraIc congruences, quadraIc reciprocity, Legendre symbol

AddiIonal topics as Ime permits selected from:

-- Zeckendorf’s Theorem

-- the Riemann zeta funcIon, Euler product, Riemann hypothesis,

-- irraIonal and transcendental numbers, Liouville numbers

-- the abc conjecture.

Required Textbook:

G. Jones and J. M. Jones, Elementary Number Theory, Springer, ISBN-3540761977 (An ebook version of this text is available.)

AddiIonal references:

(These books are not required but are nice sources of addiIonal material and perspecIves.)

J. Silverman, A Friendly IntroducIon to Number Theory, Pearson, 4th ed., 2012, ISBN-13:978- 0321816191.

A.Granville, Number Theory Revealed: An IntroducIon, Amer. Math. Soc., 2019.

K. Rosen, Elementary Number Theory, 6th ed., Addison Wesley, ISBN-13:978-0321500318.

MAT315H1S: Introduc3on to Number Theory Fall 2024

How this course is organized:

Lectures: Monday 9-10, Tuesday 9-11, room MP 202

Tutorials: Each student should be enrolled in a tutorial secIon meeIng 1 hour each week. These are scheduled for Wednesday 10-11 and 12-1, Friday 9-10

First week of classes: The first lecture will be on Tuesday, September 3th, 9-11. ***Tutorials will meet at their regular Fmes the first week.***

Here are the details, including secIon numbers and locaIons:

MAT315H                    Introduction to Number                                                               WE:10:00-11:00

1                     F            Theory                                      TUT       101                                  (UC 52)

MAT315H                    Introduction to Number                                                               WE:12:00-13:00

1                     F            Theory                                       TUT        201                                  (OI 5250)

MAT315H                    Introduction to Number                                                                FR:09:00-10:00

1                     F            Theory                                       TUT        301                                  (WO 25)

Homework: There will be weekly homework. These assignments will be posted aner the Monday lecture and will be collected on Crowdmark before the start of lecture the following Tuesday (i.e.,  due by 9:00am). Please note: No late homework will be accepted.

There will be no homework due the days of the Midterms.

Midterms: There will be two Midterm Exams,

Tuesday, October 8, 7—9pm

Tuesday, November 19, 7—9pm

The rooms will be announced later in the term.

II.               EvaluaFon/ Grading Scheme

Mark Breakdown:

Homework                      15%

Midterm Exam 1            25%

Midterm Exam 2            25%

Final Exam                      35%

Homework Assignments:

There will be 9 homework assignments, of which the two with lowest grades will be dropped.

Midterm Exams:

Dates and Imes are indicated above.

Final Assessment:

The final assessment will be held during the final assessment period in December 2024 and will be scheduled by the registrar. InformaIon about the format will be provided during the Fall semester