### Ma458 Homework 1

Ethan Bolker
Fall 2013

For Thursday's class:

1. Please fill out the questionnaire.

2. Send me email (to ` eb at cs dot umb dot edu`) from the email address you want me to use for the class mailing list.
Note: Math 458 is a writing course as well as a math course. In your other writing courses your instructors probably wanted you to use a word processor. The best language for typesetting mathematics is TeX. Learning it here will pay off in your future in mathematics.

This link is a good place to begin tex.stackexchange.com/questions/89297/prerequisites-for-using-latex-efficiently.

There you will find out that you need to have a TeX implementation and a TeX editor.

There are two good (free) TeX implementations: TeXLive (www.tug.org/texlive/ and MikTeX (http://miktex.org/.

You can use any word processing program to write TeX - even Word or Wordpad. But don't do that! Consider TeXStudio, free from http://texstudio.sourceforge.net/. For more choices, look at the wikipedia page en.wikipedia.org/wiki/Comparison_of_TeX_editors.

Here are some links that may be useful

For next Tuesday:

1. Explore as thoroughly as you can the solutions (if any) to the Diophantine equation
```	n = x2 - y2
```
(We started studying this problem in class.) In most math courses until this one, the answer to a problem is usually a computation with a circle around the conclusion, or a short algebraic proof. This course is different. Answers are complete short essays, with as much attention paid to the words as to the symbols.

If you do not understand something, or you think you do but you're not sure, say so! You won't improve your grade by faking comprehension. You will by wrestling honestly with difficult material.

Be sure to distinguish explicitly between assertions you can prove and those you suspect but can't prove. When you can prove something, try to formulate the argument in terms as elementary as possible. Some of the facts about differences of squares could be explained convincingly to a third grader who knows no algebra.

2. What is the largest number currently known to be prime? When was it discovered? How? You can find the answer on the internet: Google is a good place to begin.

3. Find some other interesting (to you) number theoretical things on the internet. You might check out number theory software, or Fermat's last theorem, or the sieve of Eratosthenes. Use your imagination. Tell me what you find (with URLs, please).

4. Read the first two chapters of Silverman (on line at A Friendly Introduction to Number Theory). Do Exercises 1.1, 1.6, 2.1, 2.4, 2.8.