Description: The first few weeks will be spent covering essentially
the material from MATH 312: divisibility, congruences, prime numbers, and
similar foundations of number theory. Some of this material might already
be familiar to you (congruences come up in courses on abstract algebra, for
instance). Once we have this foundation, there are many different directions
we could follow. Topics I would like to cover include: finding roots of polynomial
congruences; the Quadratic Reciprocity Theorem; writing numbers as sums of
squares; arithmetic functions and Dirichlet series; Farey fractions and continued
fractions; prime number estimates. I will also try to indicate the connections
between these topics and other advanced areas of number theory (algebraic
number theory, analytic number theory, diophantine approximation, etc.).
This course will not require any particular background in number theory; indeed, the course really doesn't assume anything beyond high school mathematics, with the exception of mathematical induction. What is required is “mathematical sophistication”, which certainly includes being able to understand and write proofs. I anticipate that the course will proceed at a fairly rapid pace.
Evaluation: The course mark will be based on six homework assignments (60% of the final mark) and a final exam (40% of the final mark). The homework assignments will be due every other Friday beginning September 20. Students are allowed to consult one another concerning the homework problems, but your submitted solutions must be written by you in your own words. My intention is to give a take-home, open-book final exam to be written during the final examination period; for this exam, students are not permitted to consult one another.
You can also download this course outline
in PDF format.
Correction to the final exam: problem 9(b) should refer to the equation w2+x2+y2+z2=35wxyz, not w2+x2+w2+z2=35wxyz.
The take-home final exam is now posted. Please read the instructions carefully. The due date to hand in your written solutions is Monday, December 16 by 3 pm. More information about office hours, logistics and rules for the exam, and so on is available.
Homework #6 was due on Friday, November 29.
Homework #5 was due on Friday, November 15.
Homework #4 was due on Friday, November 1.
Homework #3 was due on Friday, October 18.
Two handouts were distributed in class on Friday, October 4—come to my office if you need copies of them.
Homework #2 was due on Friday, October 4.
Homework #1 was due on Friday, September 20.