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).
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.
Binomial coefficients, binomial functions, Pascal's triangle and
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
Pascal's triangle modulo 2
The approximation of binomial(2k, k)
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