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.
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:
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