Instructor: Yuri (Yuri Burda)
E-mail: yburda (a t) math (d o t) toronto (d o t) edu
Mailbox: ``Burda'' in BA6290
Office: HU1026 (215 Huron St, 10th floor) --- make an appointment if you want to find me in the office
Office hrs: Thursday 10:30-12:30, BA6283
TAs: Lucy (Liudmyla Kadets) until Midterm, Misha (Mikhail Gudim) after midterm
Office hrs: Friday 12-2pm, BA4010, except Friday January 20. On Jan 20 the office hour will be in the lounge on the 10th floor in 215 Huron building.

Course Description

The course is an introductory course in number theory. Like number theory itself, the course is a potpourri of very different ideas and techniques applied to very different problems. The highlights of the course include discussion of Pythagorean triples, rationality of quadrics, quadratic reciprocity law, RSA cryptosystem, Pell's equation and continued fractions. Throughout the course we will emphasize connections with other parts of mathematics (even if you haven't heard of them): geometry, group and ring theory, cryptography and so on.

The prerequisites to the course include only familiarity with high-school material (performing algebraic manipulations, factoring an integer as a product of primes).

Course Format

We will meet weekly on Tuesdays from 3pm to 5pm and Thursdays 4pm to 5pm in LM 162. On Tuesdays we will be working on hard problems and proving important theorems ("developing theory"), while on Thursdays we will be solving easier problems and exercise the skill of writing down mathematical proofs. Suggested problems to work on will be posted on the course website. There will be no graded homework assignments. Quizzes will take place in the end of Thursday lecture. The time and place of term test and final exam will be announced later.

Textbook

Joseph H. Silverman, A Friendly Introduction to Number Theory, third edition.

Evaluation

Your grade will be based on three quizzes (30 minutes, 10\% each, based on suggested problems), one term test (2hrs, 25\%) and final exam (3hrs, 45\%).

There will be no make-up tests. If you miss a test or a quiz for a legitimate reason which you can document, the final exam component of your mark will be increased accordingly. The documentation must be submitted no later than 7 days after the date of the test.

Course Goals

By the end of the course you should


  • Be able to find as many Pythagorean triples as you wish
  • Solve linear Diophantine equations
  • Deal with remainders modulo an integer
  • Explain the multiplicative structure of invertible residues modulo a prime
  • State and use quadratic reciprocity law
  • Know whether a given number can be written as a sum of two squares
  • Send an encrypted message to your friend using the public-key cryptography algorithm RSA
  • Write a given quadratic irrationality as a continued fraction
  • Explain in what sense continued fractions are the best rational approximations of a given number.

Tentative Weekly Schedule

The following schedule is tentative: we might be moving faster or slower than the schedule (it is based on the schedule of the same course taught by a different instructor).

January 10, 12: Introduction of the course, Pythagorean triples, divisibility, greatest common divisor

January 17, 19: Linear Diophantine equations, congruences, Fermat's Little Theorem

January 24, 26: \textit{Quiz 1}, Euler's Phi function, Chinese Remainder Theorem

January 31, February 2: Primes, fast exponentiation

February 7, 9: RSA, primitive roots

February 14, 16: Quiz 2, Squares modulo a prime, primes as sums of two squares

February 21, 23: Reading week --- no classes

February 28, March 1: Midterm test, quadratic reciprocity

March 6, 8: Pell's equation

March 13, 15: Diophantine approximation, transcendental numbers

March 20, 22: Quiz 3, Continued fractions, Gaussian integers

March 27, 29: Unique factorization in Gaussian integers and in other rings

April 3, 5: Review, special topics

What we actually covered:
January 10: Introduction to the course, Pythagorean triples, a couple of tricks with divisibility and algebraic manipulations, solving linear diophantine equations

January 12: Rationality questions, criterion for finding all rational roots of a polynomial with integer coefficients

January 17: Greatest common divisor, Euclidean algorithm, Linear Diophantine equations compared to Euclidean algorithm
Quadratic Diophantine equations in three variables: parametrization of rational points on planar quadrics with a given rational point, examples of quadrics without rational points.

January 19: Congruences, there are infinitely many primes, there are infinitely many primes of the form 4k-1

January 24: Inverting residues, Fermat's little theorem, Euler's phi function, Chinese remainder theorem

January 28: powers modulo m and successive squaring, testing for compositeness using FLT

February 1: extracting roots mod n, RSA algorithm

Somewhere in between: a polynomial congruence of degree n has at most n solutions modulo a prime, there exists a primitive element modulo a prime

February 14: computation of 1^k+...+(p-1)^k mod p, solutions of x^k=1 mod p, the numbers b for which x^k=b mod p has solutions are exactly those for which b^m=1 mod p, where m= (p-1)/gcd(p-1,k), (-1) is a square mod p if and only if p=1 mod 4, every prime divisor of n^2+1 is of the form 4k+1, there are infinitely many primes of the form 4k+1

February 16: solving polynomial congruences modulo p^k for prime p, quiz 2

February 28: Midterm

March 1: quadratic residues, Legendre symbol, Euler's criterion, polynomial with no integer roots but with a root modulo any prime p