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.

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.

   This is really a writing course. In your other writing courses your instructors probably wanted you to use a word processor. You're welcome to try that here if you like, but writing mathematics on a computer is often difficult and time consuming. Unless you can do it easily, I'd rather you spent your time on the mathematics. So handwritten work is quite acceptable, as long as it's a clean, legible copy. Don't turn in first or even second drafts on scrap paper torn from your notebook, with with ragged edges left in place. Staple your neat pages together.

   You could consider doing the words with a word processor, leaving spaces for filling in the equations by hand on the hard copy.

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. Explore as thoroughly as you can the solutions (if any) to the Diophantine equation

   	n = x2 + y2
   

   This is a much harder problem. You probably can't get very far with it without knowing more number theory. Do the best you can. Please don't try to look up the problem somewhere. You could surely find discussions in our texts, or on line - resist the temptation!

   How far can you get with this problem from page 29 of Friedberg?

   Here is a sequence with an ordinal rule:

   	5, 17, 37, 5, 101, 5, 197, 257, 5, 401, 5,577, 
   	677, 5, 17, 5, 13, 1297, 5, 1601 ... 
   

   The mth number is the smallest factor of 4m² + 1, not counting 1 as a factor. If m is 6, then m² is 36, and 4m² + 1 is 144 + 1, or 145, and the smallest factor of 145 is 5. So the sixth number is 5. If m is 7, then 4m² + 1 comes out to be 197, and nothing goes into 197, besides 1, except 197. So the seventh number is 197.

   An inclusive rule would tell us whether any particular number, such as 3, is in the sequence or not. You cannot find this out just by using the ordinal rule and working out the sequence because of the jumps back and forth. It seems that 3 has been skipped, but maybe it will turn up later. The number 13 turned up for the first time in position seventeen, after some much larger numbers. Even if you find a thousand numbers in the sequence, with no 3, you won't know whether a 3 will turn up later. To find an inclusive rule requires deep thought. A little reflection shows that nothing besides prime numbers can be in the list, and that 2 is not in the list. If you work hard, you may see a way to prove that 3 is not in the list either. (If you can do this on your own, without knowing any number theory before, you have unusual ability.) But to find the rule that is good for all numbers is especially difficult, and we shall work on it in Chapter 7,)

   If you want to work on this problem with others in the class, or to share insights or discoveries, feel free. Just write your answer in your own words, and acknowledge other people's contributions.