# Math 406, Spring 2018: Class Announcements

## Welcome

Welcome to the class! Class announcements and useful documents will appear here. New material will be added at the top. Make it a habit to look here. I'm going to try keeping the lecture notes and an indication of the current assignments at the top of the page: watch for other things that might appear below.

## General Information and office hours

My office is Mathematics 240A. My office telephone number is 426-3011. I am teaching Math 187 and Math 406 in Spring 2018. My classes are M187 9--9:50 MWF, M406 10:30--11:45 WF. I have the regular weekly PBC meeting 2-3 Tuesday, logic seminar 3-4 Tuesday. I am proposing to have an office hour with my prospective graduate student 2-3 PM Monday, class office hours MWF 3-4 pm and TTh 10-11 am. At other times MWF from about 8:30 am until about 5 pm I'm very possibly to be found in my office. TTh I do not know what my routine will be (but I will be in in the morning since I have declared office hours then).

## The Lecture Notes from 2016 (and other tools)

Here are the lecture notes from 2016: we will transform them into "the lecture notes from 2018" as we go along. I am sure that there are some good note readers out there, who can catch typos and other slips...read, and find them (I give points if you find them before I do)!

Here is my Euclidean algorithm spreadsheet calculator.

## Final "Homework" Opportunity

Attendance at the final lecture during the final exam period (Wednesday, May 2, at 10 am) will be good for a bonus homework check. Those who do not attend will not be penalized: this is pure positive incentive. My final lectures Friday and on the second will be about two cases of Fermat's Last Theorem (sums of fourth powers cannot be fourth powers, then sums of cubes cannot be cubes).

..thinks of cookies...(doesn't promise cookies, may not remember them, but thinks of them :-)

## Test II Solutions

Here are the solutions to Test II (handwritten and scanned).

Please note that I revised the weights of the computation and proof parts in Test II; you will have received revised grades on the test and estimated course grades in individual mail.

## Test II

Test II will occur on April 20, the Friday before dead week. There will be discussion in class of exact coverage, and review materials posted here.

As a first approximation, I post links to past tests:

In Spring 2014, they did badly at the Euclidean algorithm on the first test. You all did not have that problem.

I suspect I may have provided the first question on this test as a makeup for people who did badly at the Euclidean algorithm question on the first test, as well.

Everything on these tests is fair game. I might ask about something on your test that you all did not do so well on on the first test. A variant on the last question comes to mind.

There is also stuff on this test which I could perfectly reasonably ask.

HERE is your Test II Review Sheet, which is a homework assignment due on Wednesday after the test.

HERE are your Test II Review Sheet Solutions. Final version of solutions is up, unless you all report typos or errors which need correction. There well may be some. Though I decided to provide solutions, you may still turn in your worked review sheet for homework credit.

## Second week after Spring Break

• Friday 13 April: The link for the Test II Review Sheet appears above.

Time on Wednesday will be partitioned between some progress on effectiveness of the Rabin-Miller test (which I will do first, to make sure it actually happens) and test review: best prepared for by studying the sample tests and the review sheet!

• Wednesday 11 April: In keeping with the exercise in class, I'll make the exercise delightfully informal. Choose three primes of the form 4k+1, each larger than the last example 1549 done in class, and carry out the algorithm demonstrated in class (and found on p. 187 of the book) to find an expression for each of them as a sum of two squares. This will force me to build my own Python code to check your work, as between you all you will no doubt choose many different primes! Any student who writes their own Python code to execute this task may submit it for additional credit (one of you has submitted code to compute Legendre symbols, to which similar remarks apply).

This is due next Wednesday. It's very likely that I will come up with another homework set (possibly a set of problems for test review, possibly new stuff) on Friday.

## For the week after Spring Break

• Writing on 3/28/18: I have posted exercises as section 23 of the notes, notionally due on April 6th. These are all about quadratic reciprocity in one way or another, and they do have some new ideas in them which you should be exploring. I will be more likely to say something helpful about them if you have been working on them and Ask Me About Them... I did survive the Wedding (it was not in the least like the one in Game of Thrones)!

## Week 9

• Friday March 9: Your homework is described on page 46 of the notes. There is some cryptographic stuff with a flexible due date and some problems from the book on Mersenne prime/perfect number issues and primitive root issues which are listed in the notes. I fixed the factorization problem in the notes on the Primitive Root Theorem. Please read that proof carefully and tell me about any problems you have with it. I owe you a subsection on the ElGamal system for the notes, and also functions in the Sage project to turn text messages to numbers and vice versa; I'll try to attend to both of these matters during the weekend. I hope there will be lots of lively questions on Wednesday (if not before!)

## Week Eight?

• Wednesday, 28 Feb: Note that you have an extension on the homework until next Wednesday. Come armed with questions to the Friday class. Also, you are assigned the exercise of completing problem 7 on the test based on my remarks in class for the coming Friday class. If you weren't in class today, Ill accept this later. On the Sage front, please make your own copy of my RSA files and encrypt and decrypt a message to yourself and make sure it works.

Problem 7, if you do not recall, was this. Show that in a PPT (a,b,c), c cannot be divisible by 7. In addition, state a pattern in the behavior of the remainders on division by 7 of a and b in PPT's if you can see one. The secret is to use calculations of all possible values of the squares of a and b in mod 7 arithmetic (suitably, of course). If you want to explore and see if there is something to say about PPT's in mod 11, do so...I have no idea whether there is anything interesting about them.

## Week Six

• Friday 16 Feb: Problems listed in section 13 in the notes (from chapters 17, 18, 19 in the book) are due the Friday after the test (UPDATE: I have given an extension to the Wednesday after the stated Friday; please come to the Friday class armed with your Questions). I have more to say about some of these topics, and I'll be giving computer demos.

## Week Five

• Friday February 9: Finished talking about the Chinese remainder theorem and the method of computing the Euler phi function.

Posted homework (on Sunday, sorry I'm slow): do the problems from the book listed in section 9.4 of the notes (p. 36). Problems are noted to be optional only individually! A lot of this is computational.

## Week Four

• Friday, February 2: Homework 3 will still be accepted next Wednesday due to confusion about which problems were optional (only two of them!).

Homework 4, due next Friday: do the exercises listed in section 8.3 of the notes (exercises from chapter 8 in the book).

## Week Three

• Friday, January 26: Homework 3, due next Friday: do the exercises in section 6.4 of the notes (these are problems from the book) Get started before Wednesday (don't slack off when you finish the previous assignment for Wed)...as you may have questions for me...

## Week Two

• Friday, January 19: Homework 2, due next Friday: do the exercises in section 4.3 of the notes (these are problems from the book).

Since this was not posted correctly, it is now due Wednesday the 31st rather than Friday the 26th. I really had edited the file correctly on Friday; I must have dropped it in the wrong directory. I would appreciate email if the homework doesn't appear by Friday at 6 pm or so (there might be factors which delay me) and certainly if it is not there on Saturday morning! There will be homework assigned on Friday the 26th, which will be due as usual the following Friday.

(Thanks to a student): Should this be the answer key to problem 5.5?

On a more serious note, you might like my Python functions in this file for explorations of the process in 5.5 (the guess that this process always terminates is called the Collatz Conjecture).

## Week One

• Friday, January 12: Homework 1, due Friday January 19th: do the exercises on page 9. The last one is a challenge problem, and you should not worry unduly if you are not successful at it. Do be sure you are looking at the updated version of the file: if it has an exercise about the div and mod operations, it is updated.

## Test I--February 23

Here are some practice exams. Be aware that you need to practice techniques we have talked about in class with a simple non-graphing calculator. I strongly recommend practice with the tabular format for computing gcd's and with the repeated squaring technique for computing powers in modular arithmetic.

This should give a very clear impression of what you need to know for the exam. I will probably have a little more to say on Wednesday.