- Sections 1.2–1.3: divisibility, gcds, primes
- Sections 2.1–2.3 and 2.6–2.8: congruences, solutions of congruences, Chinese remainder theorem, prime power modulus, prime modulus, primitive roots and power residues
- Sections 3.1–3.6: quadratic residues, quadratic reciprocity, Jacobi symbol, sums of two and four squares
- (if time permits: binary quadratic forms)
- Sections 4.2–4.4: arithmetic functions, Möbius inversion, recurrence functions
- (if time permits: A.4, linear recurrences)
- Sections 5.1 and 5.3: solving
*ax*+*by*=*c*in integers, Pythagorean triangles - (if time permits: 5.2 and 5.4, simultaneous linear equations and assorted examples)
- Sections 6.1–6.3: Farey sequences, rational approximations, irrational numbers (as time permits)
- Sections 7.1–7.8: the Euclidean algorithm, uniqueness, infinite continued fractions, (best possible) approximations of irrationals by rationals, periodic continued fractions, Pell's equation (as time permits)
MATH 437 is an honours course! It treats roughly the same material as MATH 312 and 313 combined, and will take nearly twice as much work as either of those classes. Note that a student cannot have credit for both MATH 312 and MATH 437, nor for both MATH 313 and MATH 437. To enroll in MATH 437, an undergraduate student must have already taken, or be taking simultaneously, one of MATH 320 or MATH 322.
The word “elementary” in the title does not mean the course isn't difficult; rather it means that the course doesn't use techniques from real or complex analysis or from abstract algebra. The course will not require any particular background in number theory. What is required is “mathematical sophistication”, which certainly includes being able to understand and write proofs. Be forewarned that this course will be taught at the level of a graduate course. Honours students typically will be well-equipped to succeed in this course.
Your homework will be marked on correctness, completeness, rigor, and elegance. A correct answer will not earn full marks unless it is completely justified, in a rigorous manner, and written in a logical sequence that is easy to follow and confirm. I plan on being pedantic about completeness of solutions (for example, if you invoke Euler's theorem to assert that
You are very welcome to come by my office hours and ask questions about the lecture material, homework/discussion problems, clarity of style in proof writing, or related mathematical content. If office hours conflict with your schedule, you may make an appointment with me via email; you are also welcome to just drop by my office and see if I'm there—about half the time, I'll be free to talk right then. Students are allowed to consult one another concerning the homework problems, but
Unless otherwise announced, the in-class group problem sessions will take place the lecture after each assignment is due—so Wednesday, September 16; Wednesday, September 23, and so on through Monday, November 30. (Hopefully there will be no exceptions to this rule, so that you can easily predict when the group problem sessions will be.) | ||