User:Marty/UCSC Math 110 Fall 2008

The final exam will take place on December 9, at 7:30 pm.

Here is a final study guide to help with your studies.

Filix will be holding a review session, on Tuesday, December 9, from 2:30pm - 4:00 pm, in J.B. Engineering, Room 360.

Here are copies of the two quizzes, and one modular arithmetic worksheet from the course. Click on the links below, then click on the "blah.pdf" file to get the file:
 * [[Image:Math110Quiz1.pdf‎]]
 * [[Image:Math110Quiz2.pdf]]
 * [[Image:Math110ModularQuiz2.pdf]]

There will be office hours on Monday (with Marty) from 1:30 pm until 3:30 pm.

=Syllabus=

Introduction
Welcome to Math 110, an introduction to number theory. In this class, we will explore the arithmetic properties of integers. Fundamental questions about integers are often best approached through techniques of geometry, combinatorics, and analysis. We will utilize all of these approaches in our study of integers. The course will be administered on the new SlugMath wiki (!) at. You can access the page for this class, by navigating to the "Classes" section (using the sidebar), or directly at the url:

Getting Help
For help with this wiki, click on the help link, on the left sidebar. Office hours will be held at the following times in J. Baskin Engineering, Room 361.
 * Tuesday: 11:00am - 12:00 pm.
 * Thursday: 4:00pm - 5:00 pm.

If these times do not work, feel free to contact Marty to set up an appointment. To see Marty's calendar, you can go to his home page here. You can always e-mail Marty or Filix with questions about the course.
 * Marty can be contacted at weissman AT ucsc DOT edu
 * Filix can be contacted at fmaisch AT ucsc DOT edu

Homeworks
There will be 9 homework assignments due in this class, of which you may drop the lowest 2 scores. You should not plan to skip any homework assignments -- the dropped scores are meant to account for emergencies and such. With that in mind, no extensions on homeworks will ever be granted (except in the most exceptional circumstances with the professor's permission), since you get to drop the lowest two scores.

Homework assignments are due by the beginning of class on their due date (which is usually Wednesday). They may be turned in anytime before the beginning of class, by sliding them under the door of my office (JBE 361B). Of course, they may be submitted in the classroom as well! Beginning with the third (probably) homework assignment, I expect assignments to be typeset and printed using a LaTex package. Further instructions will be given in the first few weeks, to assist you in this process.

Quizzes and Exams
There will be two short (30-minute) quizzes, one longer (1-hour) test, and one longest (2-3 hour) final exam.

The final exam will take place on December 9, at 7:30 pm.

Grading
Recall that there are 9 homework assignments, of which the best 7 will count towards your grade. Altogether, homework assignments will account for 50 percent of your grade. Each 20-minute quiz will account for 5 percent of your grade. Altogether, quizzes will account for 10 percent of your grade. The longer, 1-hour test will account for 10 percent of your grade. The final exam will account for 30 percent of your grade. Observe that $50 + (2 \times 5) + 10 + 30 = 100$.

In the end, a letter grade will be assigned based upon your numerical score as computed above. A curve will be used, based upon student performance and my expectations. Generally, the average grade in my classes of this type is around a B or B+ -- I expect this class to be no different.

Honesty Policy
All students are expected to follow basic academic honesty policies, as determined by UCSC. Students are encouraged to work in groups on the homework assignments. However, every student should turn in their own, distinct, individual work -- students cannot simply copy the work of their peers under any circumstances. All quizzes, tests, exams are given to individuals, and group work is not permitted. All violations will be reported to the appropriate academic authorities.

Attendance Policy
Attendance in class is very highly recommended. There really is no substitute, though notes are posted online. With that in mind, attendance in class and section will not be taken. Please do not show up to class more than 5 minutes late. If tardiness is repeated, you will be gently tossed from the class for a day. It is disruptive and annoying to come in late on a regular basis.

=Resources=

Assumed Knowledge
For this class, it is assumed that you have a knowledge of basic facts about arithmetic, natural numbers, integers, rational numbers, and real numbers. It is also assumed that you have some experience in reading and writing proofs, including direct proofs, inductive proofs, and proofs by contradiction.

Here is a compendium of facts about numbers, which you may assume in any proofs you write in this class.

Books
The two textbooks for the class are the following:
 * The sensual quadratic form, by John H. Conway.
 * Elements of number theory, by I.M. Vinogradov.

These books are very different in style and substance. We will be covering much of the material from Vinogradov's book, but without as much emphasis on analytic techniques (Vinogradov was most famous for his applications of analytic methods to problems in number theory).

We will be thoroughly covering the first chapter of Conway's book, and perhaps some of the fourth chapter as well. Conway is a mathematician who's work spans diverse fields of mathematics; the first chapter of his book describes a visual approach to the classical theory of binary quadratic forms.

LaTex
Here is the not so short introduction to LaTex The chapters on typesetting text and mathematics are particularly useful!

Latex has been installed at Kresge
Here are instructions from the IT people, on using MikTex and TexMaker at the Kresge computer lab:

To run the program after logon do the following: the Desktop)
 * 1) Select the Start Menu (or double-click on the "Applications" folder on
 * 1) Double-click on the "Class folders" folder
 * 2) Double-click on "Math" folder
 * 3) Double-click on "TexMaker" folder
 * 4) Double-click on "TexMaker" icon

The program is run from the server, however the server is read-only. So any data files created by users will need to be saved to the user's UCSC Home Directory which is mapped as "X:" on the PCs, or USB Flash Drive, or CD-R/RW. Let me know if you have any questions. Temporary files can be created on the Desktop or C:\Usertemp.

Outside Resources
It is interesting to look at Euclid's Elements (especially Book VII), to see the ancient perspective. An online edition, in Greek and English, can be found here. It is helpful, when reading the Elements, to understand the following:
 * Numbers are almost always given as lengths of line segments (e.g., the number $AB$ denotes the distance from point $A$ to point $B$).
 * The word "measures" corresponds to our word "divides". For example, the number $AB$ measures the number $CD$, if the line segment $CD$ can be broken up into a finite number of segments, each congruent to $AB$.

You might enjoy poking around the ArXiv to see the new math articles of the day.

You might wish to look at some of Euler's work (in Latin). You can find some of this work at the Euler Archive. Especially, check out E271, around page 81, for a description of the "totient" and the Fermat-Euler theorem.

Gauss's Disquisitiones Arithmeticae can be found online too.

Online Computation
The MAGMA calculator can be found at: It can be used for incredible computations in number theory. Some of the documentation can be found at

A calculator can be found online, which carries out division with remainder for polynomials. It is found at:

Garrett has a modular exponentiation calculator at:

Google search has a rudimentary calculator, which can do some small modular arithmetic. For example, try typing "(23 + 35) mod 7" into your Google search bar! Don't try any large exponents -- Google is not that smart... yet.

This wiki
This wiki will be a useful resource, and you are encouraged to use it to understand the "big picture" of mathematics, and for this class.

To see the topics covered in the course (so far), go to the week-by-week tab, and look at the links.

=Week-by-Week=

Week 0

 * Day(s): Friday, Sept. 26.
 * Admin: Review the syllabus.  Wiki usage.  LaTex.
 * Math:
 * Lecture notes: For  week 0

Week 1

 * Day(s): MWF, Sept. 29, Oct 1,3.
 * Admin: Homework 1, due Wednesday, Oct. 1.  Read Vinogradov, Sections I.1, I.2, I.5, I.6.
 * Math:
 * Lecture notes: For  week 1

Week 2

 * Day(s): MWF October 6,8,10
 * Admin: Homework 2 due October 8.
 * Math:
 * Lecture notes: For  week 2

Week 3

 * Day(s): MWF, Oct. 13,15,17.  No office hours on Thursday.  Wednesday 5-6 instead!
 * Admin:  Homework 3 due Oct. 15.  Short quiz Oct. 17.  Study the main definitions and statements of  basic number theory, practice using the Euclidean algorithm to find GCD and solve linear Diophantine equations.  Make sure that you can do homework problems.  Know definitions of prime, GCD, LCM, very precisely!  Quiz will not cover lax vectors, topographs, etc.. from this week.
 * Math:
 * Lecture notes: For  week 3

Week 4

 * Day(s): MWF Oct. 20,22,24
 * Admin:  Homework 4 due Oct. 22.
 * Math:
 * Lecture notes: For  week 4

Week 5

 * Day(s): MWF Oct. 27,29,31.
 * Admin:  Homework 5 due Wednesday, Oct. 29.  Halloween!
 * Math:
 * Lecture notes: For  week 5

Week 6

 * Day(s): MWF Nov. 3,5,7
 * Admin: Homework 6 due on Wednesday.  Test on Friday, November 7.  Here's a study guide
 * Math:
 * Lecture notes: For  week 6

Week 7

 * Day(s): MWF, Nov. 10,12,14.
 * Admin: Homework 7 due Wednesday, Nov. 12.
 * Math:
 * Lecture notes: Check out the following lecture notes for  week 7.  Look at Chapters III and the beginning of IV, in Vinogradov.

Week 8

 * Day(s): MWF, Nov. 17,19,21.
 * Admin: Homework 8 due Wed., Nov. 19.
 * Math:
 * Lecture notes: For  week 8

Week 9

 * Day(s): MW, Nov. 24,26.
 * Admin: Thanksgiving on Nov. 27!.  Quiz on November 24!  No Homework Due.
 * Math: Counting Turkeys.
 * Lecture notes: For  week 9

Week 10

 * Day(s): MWF, Dec. 1,3,5.
 * Admin: Homework 9 due Fri., Dec. 5.
 * Math: Quadratic Reciprocity, computations and proofs!
 * Lecture notes: For  week 10