Number Theory            Last updated: March 21, 2009.
MATH 346 and 377. Winter 2009.

Time and Place: MWF 11:35 - 12:25, BURN 306.

Text Book: An Invitation to Modern Number Theory by Steven J. Miller and Ramin Takloo-Bighash.
Course prerequisites: MATH 235 and a lot of enthusiasm and willingness to think 'long and hard'.
Method of Evaluation: 20% midterm, 75% final exam, 5% assignments. There will be 9 assignments. The grade for each assignment will be either G or B. To get credit for the assignments you have to get at least 7 G, else you get zero. A non-programmable pocket calculator (battery operated) is required for the final exam.

Tutorial hour and make-up hour: Friday 14:30 - 15:30 in BURN 1B45. Two such meetings will be make-up classes, the rest is a tutorial hour I am giving voluntarily and attendance is optional.
Office hours: Monday 10:30 - 11:30, Friday 10:30 - 11:30.
Special Office Hours prior to the exam: the usual office hours are cancelled. Here are the office hours prior to the exam:
Monday, April 20, 11:30 - 12:30.
Wednesday, April 22, 09:00 - 10:00.
Thursday, April 23, 09:00 - 10:00.
Monday, April 27, 11:00 - 13:00.

Statement on Acadmic integrity: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see for more information).

The calendar description is: Divisibility. Congruences. Quadratic reciprocity. Diophantine equations. Arithmetical functions.
We shall cover those and additional topics. Here are the topics I hope to cover, time allowing.

1. Binomial coefficients, binomial functions, Pascal's triangle and polynomial arithmetic functions. Sum of k-th powers.
2. Congruences. Chinese Remainder Theorem. Quadratic reciprocity. Euler's function. Mobius inversion and multiplicative arithmetic functions. Perfect numbers. Carmichael numbers.
3. Several proofs of infinitely many primes. The prime number theorem. Chebyshev's theorem and Bertrand's postulate. Primes of the form x^2 + y^2.
4. Diophantine approximations. Dirichlet's and Liouville's theorems. Some transcendental numbers. e is irrational; e^r and pi^2 are irrational.
5. Continued fractions. Some famous continued fractions. Optimal approximations. Pell's equation and Archimedes' cattle problem. RSA and security of RSA decription key.
6. Counting solutions to polynomial equations in finite fields. The zeta function. Quadratic equations. Fermat's hyperplanes. Gauss and Jacobi sums.

Additional text books:

  1. Hardy and Wright. An introduction to the theory of numbers.
  2. Silverman. A friendly introduction to number theory.
  3. Baker. A concise introduction to number theory.
  4. Ireland and Rosen. A classical introduction to modern number theory.
  5. Rosen. Elementary number theory and its applications.
  6. Everest and Ward. An introduction to number theory.

Assignments: Hand out assignments during Monday morning classes, or by noon at the main office.
Assignment 1        
Assignment 2        
Assignment 3        
Assignment 4       
Assignment 5        
Assignment 6        
Assignment 7        
Assignment 8        
Assignment 9       

Pascal's triangle
Pascal's triangle modulo 2
The approximation of binomial(2k, k)
Fibonacci squares
Some example of PARI (Euler's phi function).
PARI GP can be downloaded here.
An example of a PARI session concerning primality testing and factoring.
Approximation of pi(x) by Li(x)
Approximation of pi(x) by Li(x) - Li(x^(1/2))/2
Archimedes's cattle problem