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

• M:
  • Read: Chapter 10
  • Submit T&Q17 on Chapter 10
• W:
  • Read: Chapter 11
  • due today:
    • some contribution to the wiki
    • HW8:
      1. Give two proofs of the following statement: "If $p$ is prime then $k^p\equiv k\mod{p}\ \forall k\in\ZZ$", as follows
         1. A very easy one based on Fermat's Little Theorem.
         2. A slighty more computational one, which uses the binomial theorem and a careful analysis of the binomial coefficients $\begin{pmatrix} p\\j\end{pmatrix}$  when $p$ is prime and $1\le j\le p-1$.
      2. Prove the statement "$\forall n\in\ZZ$ if $n$ is not a multiple of $17$, then either $n^8+1$ or $n^8-1$ is divisible by $17$."
      3. problem 10.1 in the textbook
  • Submit T&Q18 on Chapter 11
  • please contribution to the wiki!
• F:
  • Read: Chapter 11
  • Submit T&Q19 on Chapter 11
  • please contribution to the wiki!

• M:
  • Read: Chapter 12
  • Submit T&Q20 on Chapter 12
  • due today: HW9 = {11.3, 11.9}
  • please contribution to the wiki!
• W:
  • Read: Chapter 13
  • Submit T&Q21 on Chapter 13
  • please contribution to the wiki!
• F:
  • Read: Chapter 14
  • Submit T&Q22 on Chapter 14
  • please contribution to the wiki!
  • start HW10, which will be due on Monday
  • Today [Friday] is the last day to withdraw (with a W) from classes

• M:
  • Read: Chapter 15
  • Submit T&Q23 on Chapter 15
  • the wiki!
  • due today: HW10 = {12.2, 13.3}
• W:
  • Read: Chapter 16
  • Submit T&Q23 on Chapter 16
  • the wiki!
• F:
  • Read: Chapter 17
  • Submit T&Q24 on Chapter 17
  • the wiki!
  • due today: HW11 = {14.1, 15.3}
  • Start problem 15.1, it will be due after Spring Break!

• M:
  • due today: HW12 = {15.1, 17.3}
  • review for Midterm II; see this review sheet
• W:
  • more review for Midterm II
  • please have handed in all outstanding HW assignments and be ready to discuss those (and other) problems
• F:
  • Midterm II

• M:
  • going over Midterm II
• W:
  • Read: Chapter 18
  • Submit T&Q24 on Chapter 18
  • the wiki!
  • hand in Midterm II revisions, if you like
• F:
  • Read: Chapter 18
  • We'll discuss Chapter 19 in class -- please read it this weekend
  • Submit T&Q24 on Chapter 18
• Weekend:
  [yes, the weekend]
  • Read: Chapter 19 and this Wikipedia article
  • Submit T&Q25 on Chapter 19

• Weekend:
  [repeating from above, in case someone missed it]
  • Read: Chapter 19 and this Wikipedia article
  • Submit T&Q25 on Chapter 19
  • start HW13, which is due Monday
• M:
  • Read: Chapter 21
  • Submit T&Q26 on Chapter 21
  • the wiki!
  • due today: HW13 = whichever two you like out of {18.1, 18.2, 19.1, 19.4}
  • last day to hand in Midterm II revisions, if you like
• W:
  • Read: Chapter 21 and Chapter 22
  • Submit T&Q27 on Chapter 21 or Chapter 22
  • the wiki!
• F:
  • Read: Chapter 22
  • Submit T&Q28 on Chapter 22
  • the wiki!

• M:
  • Read: Chapter 23
  • Submit T&Q29 on Chapter 23
  • the wiki!
  • due today: HW14 = {21.1, 21.8, 22.6}
• W:
  • Read: Chapter 24
  • Submit T&Q30 on Chapter 24
  • the wiki!
• F:
  • Read: Chapter 25
  • Submit T&Q31 on Chapter 25
  • the wiki!

• Exam week, no classes.
• Wednesday at 3pm in PM213: review for the final exam; see also this review sheet
• Our FINAL EXAM is scheduled for Friday, May 4th, 10:30-12:50 in our usual classroom.