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:*- bureaucracy and introductions
- what would make a good
**T&Q** - some basic sets of numbers
- natural numbers, $\NN$
- integers, $\ZZ$
- rational numbers, $\QQ$

- what is Number Theory?
- 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:*- how to submit electronic
**HW**s,**CCT**sections, and**T&Q**s - some basic terminology and notation:
- logical and basic set theoretic terminology/notation
- some basic sets of numbers
- primes/composites
- triangular numbers
- square/cube/higher power
- perfect numbers
- the Fibonacci sequence
- real numbers $p$-adic numbers?)

- 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

- good careful defintions and theorem [lemma, propostion...]
statements
- looking over instructor's
**CCT**entry for*Chapter 1*

- looking over instructor's
- good careful proofs
- some proof strategies:
- direct verification
*["unpacking"]* - contradiction
- induction

- direct verification

- some proof strategies:

- how to submit electronic
**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*:- Your name.
- 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.)
- Your year/program/major at CSUP.
- The reason you are taking this course.
- What you intend to do after CSUP, in so far as you have an idea.
- Past math classes you've had.
- Other math and science classes you are taking this term, and others you intend to take in coming terms.
- Your favorite mathematical subject.
- Your favorite mathematical result/theorem/technique/example/problem.
- Anything else you think I should know (disabilities, employment
or other things that take a lot of time,
*etc.*) - [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*,**SCL**s are:- Zack
- Zeb

*Content:**Pythagorean triples*- a
**primitive**Pythagorean triple **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:**W:***Read:***Chapter 3*,**SCL**s are:- Angela
- Glen

*Content:*- rational equations versus integral equations
- working through the proof of Theorem 3.1 quite carefully
- 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.
- notice a connection between a geometric and a suitably defined number theoretic problem!

**Submit T&Q4**on*Chapter 3***due today:****SCL**s:**CCT***Chapter 2*- non
**SCL**s:**HW2**= {2.1, 2.2}

*F:***Read:***Chapter 3*,**SCL**s are still- Angela
- Glen

*Content:*- comments on
**HW1** - discussion of problems 3.1, 3.2 (also 2.7 and 2.8, time permitting; for 2.8, see this page)

- comments on
**Submit T&Q5**on*Chapter 3*

*M:***Read:***Chapter 4*, no**SCL**s for this chapter*Content:*- some history
- discussion of problem 4.2

**Submit T&Q6**on*Chapter 4***due today:****SCL**s:**CCT***Chapter 3*- non
**SCL**s:**HW3**= {3.3, 3.4}

**Today [Monday] is the last day to drop classes without a grade being recorded.**

*W:***Read:***Chapter 5*,**SCL**s are:- Sabrina
- Brandon

*Content:*- discussion of the Division Algorith, remainders,
*etc.* - starting the proof that the Euclidean algorithm works

- discussion of the Division Algorith, remainders,
**Submit T&Q7**on*Chapter 5*

*F:*

*M:**Content:*- discussion of HW problem 5.3

**Submit T&Q9**on*Chapter 6*

*W:***Read:***Chapter 6*,**SCL**s are- the double-Z team

*Content:*- 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 $a,b\in\ZZ$.
- a moment of realization of how simple was the fact that the smallest positive element of $(a,b)$ must be $\gcd(a,b)$.
- going over a slightly different version of induction, for whic see a summary at the wiki
- 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:****SCL**s:**CCT***Chapter 5*- non
**SCL**s:**HW4**= {5.3, 5.4}

*F:***Read:***Chapter 6*,**SCL**s are still- Zack
- Zeb

*Content:*- a different approach to the $\gcd$ argument than appears in the book; see the wiki for much more info

*M:***Read:***Chapter 7*,**SCL**s are:- ...in a drammatic popular uprising,
**we are all the**— please check out the the wiki and make sure you add*SCL*s this weekby next Monday*something*

- ...in a drammatic popular uprising,
*Content:*- largely the theorem about when primes divide a product

**Submit T&Q11**on*Chapter 7***due today:****SCL**s:**CCT***Chapter 6*- non
**SCL**s:**HW5**= {6.4, 6.5}

*W:***Read:***Chapter 7*,**SCL**s are still- ... we, the people ...

*Content:*- proving the
*Fundamental Theorem of Arithmetic*

- proving the
**Submit T&Q12**on*Chapter 7*

*F:***Read:***Chapter 8*,**SCL**s are:- ...still the proletarian masses...

*Content:***Submit T&Q13**on*Chapter 8*

*M:***Read:***Chapter 8*,**SCL**s are:- ...still the huddled masses, yearning to be free...

*Content:***Submit T&Q14**on*Chapter 8***due today:**- some contribution to
**CCT***Chapter 7*(from**everyone**since there are no particular**SCL**s at the moment) **HW6**= {7.1, 7.2, 7.5} (from**everyone**)

- some contribution to

*W:*- Review for
**Midterm I** **due today:**- some contribution to
**CCT***Chapter 8*(from**everyone**!) **HW7**= {8.1, 8.2} (from**everyone**)

- some contribution to

- Review for
*F:***Midterm I, In-class part**

*M:***Midterm I**- going over
**Midterm I**

*W:***Read:***Chapter 9*- hand in
**Midterm I**revisions, if you like **Submit T&Q15**on*Chapter 9*

*F:***Read:***Chapter 9***Submit T&Q16**on*Chapter 9*

*M:***Read:***Chapter 10***Submit T&Q17**on*Chapter 10*

*W:***Read:***Chapter 11***due today:**- some contribution to the wiki
**HW8**:- Give
**two**proofs of the following statement:*"If $p$ is prime then $k^p\equiv k\mod{p}\ \forall k\in\ZZ$"*, as follows- A very easy one based on
*Fermat's Little Theorem*. - 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 $1\le j\le p-1$.

- A very easy one based on
- 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 $17$."* - problem 10.1 in the textbook

- Give

**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:**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!

**Spring Break!**No classes, of course.

*M:**W:*- more review for
**Midterm II** - please have handed in all outstanding
**HW**assignments and be ready to discuss those (and other) problems

- more review for
*F:***Midterm II**

*M:*- going over
**Midterm II**

- going over
*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
is scheduled for*FINAL EXAM***Friday, May 4th, 10:30-12:50 in our usual classroom**.