Colorado State University, Pueblo; Spring 2012
Math 419 — Number Theory
Course Schedule & Homework Assignments

Shortcuts:

• Wiki is a link to the class wiki (which is open only to registered students in the class).
• Syllabus is a link back to the course syllabus/policy page.

This schedule is will be changing very frequently, please check it at least every class day, and before starting work on any assignment (in case the content of the assignment has changed).

• M:
  • Read: Introduction [p. 1-2] and Chapter 1
  • Content:
    1. bureaucracy and introductions
    2. what would make a good T&Q
    3. some basic sets of numbers
       1. natural numbers, $\NN$
       2. integers, $\ZZ$
       3. rational numbers, $\QQ$
    4. what is Number Theory?
    5. some good, basic Number Theoretic problems:
       • existence of Pythagorean Triples
       • "Fermat's Last Theorem"
       • Twin Primes
• W:
  • Read [before class, as always!]: Chapter 1
  • Content:
    1. how to submit electronic HWs, CCT sections, and T&Qs
    2. some basic terminology and notation:
       • logical and basic set theoretic terminology/notation
       • some basic sets of numbers
         1. primes/composites
         2. triangular numbers
         3. square/cube/higher power
         4. perfect numbers
         5. the Fibonacci sequence
         6. real numbers $p$-adic numbers?)
    3. more on basic Number Theoretic problems
       • the number of primes, or of primes in particular sets
       • existence of numbers of various "shapes"
       • some Chapter 1 problems
    4. good careful defintions and theorem [lemma, propostion...] statements
       • looking over instructor's CCT entry for Chapter 1
    5. good careful proofs
       • some proof strategies:
         1. direct verification ["unpacking"]
         2. contradiction
         3. induction
  • Submit T&Q1 on Chapter 1 [at least an hour before class, as always!]
  • Do HW0: Send me e-mail (to jonathan.poritz@gmail.com) telling me:
    1. Your name.
    2. Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)
    3. Your year/program/major at CSUP.
    4. The reason you are taking this course.
    5. What you intend to do after CSUP, in so far as you have an idea.
    6. Past math classes you've had.
    7. Other math and science classes you are taking this term, and others you intend to take in coming terms.
    8. Your favorite mathematical subject.
    9. Your favorite mathematical result/theorem/technique/example/problem.
    10. Anything else you think I should know (disabilities, employment or other things that take a lot of time, etc.)
    17. [Optional:] If you were going to be trapped on a desert island alone for ten years, what music would you like to have?
• F:
  • Read: Chapter 2, SCLs are:
    • Zack
    • Zeb
  • Content:
    1. Pythagorean triples
    2. a primitive Pythagorean triple
    3. The Pythagorean Triples Theorem
       • statement
       • ideas
       • proof
  • Submit T&Q2 on Chapter 2
  • Hand in HW1: 1.2, 1.3
  • Today [Friday] is the last day to add classes.

• M:
  • Read: Chapter 2, SCLs are still:
    • Zack
    • Zeb
  • Content:
    1. comments on HW1
    2. more on The Pythagorean Triples Theorem
    3. see also an alternate proof
  • Submit T&Q3 on Chapter 2
• W:
  • Read: Chapter 3, SCLs are:
    • Angela
    • Glen
  • Content:
    1. rational equations versus integral equations
    2. working through the proof of Theorem 3.1 quite carefully
    3. relationship of the proof to the idea of parameterizing the circle in an unusual way — not by angle, which would be hard to turn into a condition which measures when that point on the circle has rational coordinates, but by drawing a line and seeing where it intersects the circle. This is like the process of stereographic projection, which is used to identify the sphere with the plane.
    4. notice a connection between a geometric and a suitably defined number theoretic problem!
  • Submit T&Q4 on Chapter 3
  • due today:
    • SCLs: CCT Chapter 2
    • nonSCLs: HW2 = {2.1, 2.2}
• F:
  • Read: Chapter 3, SCLs are still
    • Angela
    • Glen
  • Content:
    1. comments on HW1
    2. discussion of problems 3.1, 3.2 (also 2.7 and 2.8, time permitting; for 2.8, see this page)
  • Submit T&Q5 on Chapter 3

• M:
  • Read: Chapter 4, no SCLs for this chapter
  • Content:
    1. some history
    2. discussion of problem 4.2
  • Submit T&Q6 on Chapter 4
  • due today:
    • SCLs: CCT Chapter 3
    • nonSCLs: HW3 = {3.3, 3.4}
  • Today [Monday] is the last day to drop classes without a grade being recorded.
• W:
  • Read: Chapter 5, SCLs are:
    • Sabrina
    • Brandon
  • Content:
    1. discussion of the Division Algorith, remainders, etc.
    2. starting the proof that the Euclidean algorithm works
  • Submit T&Q7 on Chapter 5
• F:
  • Read: Chapter 5, SCLs are still
    • Sabrina
    • Brandon
  • Content:
    1. more on the proof that the Euclidean Algorithm works
    2. introduction to our wiki
  • Submit T&Q8 on Chapter 5

• M:
  • Content:
    1. discussion of HW problem 5.3
  • Submit T&Q9 on Chapter 6
• W:
  • Read: Chapter 6, SCLs are
    • the double-Z team
  • Content:
    1. defined the notation $(n)=\{an | a\in\ZZ\}\subset\ZZ$ for the set of multiples of a given $n\in\ZZ$; likewise, $(n,m)=\{an+bm | a,b\in\ZZ\}\subset\ZZ$ is the set of integer linear combinations of two given $a,b\in\ZZ$.
    2. a moment of realization of how simple was the fact that the smallest positive element of $(a,b)$ must be $\gcd(a,b)$.
    3. going over a slightly different version of induction, for whic see a summary at the wiki
    4. the construction in the book of the coefficients $x$ and $y$ for which $ax+by=\gcd(a,b)$ is usually called the Extended Euclidean Algorithm
  • Submit T&Q10 on Chapter 6
  • due today:
    • SCLs: CCT Chapter 5
    • nonSCLs: HW4 = {5.3, 5.4}
• F:
  • Read: Chapter 6, SCLs are still
    • Zack
    • Zeb
  • Content:
    1. a different approach to the $\gcd$ argument than appears in the book; see the wiki for much more info

• M:
  • Read: Chapter 7, SCLs are:
    • ...in a drammatic popular uprising, we are all the SCLs this week — please check out the the wiki and make sure you add something by next Monday
  • Content:
    1. largely the theorem about when primes divide a product
  • Submit T&Q11 on Chapter 7
  • due today:
    • SCLs: CCT Chapter 6
    • nonSCLs: HW5 = {6.4, 6.5}
• W:
  • Read: Chapter 7, SCLs are still
    • ... we, the people ...
  • Content:
    1. proving the Fundamental Theorem of Arithmetic
  • Submit T&Q12 on Chapter 7
• F:
  • Read: Chapter 8, SCLs are:
    • ...still the proletarian masses...
  • Content:
    1.  
  • Submit T&Q13 on Chapter 8