**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.**

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 www.mcgill.ca/integrity for
more
information).

__
Syllabus__:

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__:

- Hardy
and Wright. An introduction to the theory of numbers.
- Silverman.
A friendly introduction to number theory.
- Baker.
A concise introduction to number theory.
- Ireland
and Rosen. A classical introduction to modern number theory.
- Rosen.
Elementary number theory and its applications.
- 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

Handouts:

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