I've asked for a lot in the questions that follow. I don't know how much. If you find that you are spending more time on this course than is reasonable given the shape of your life, skip some of it. But try to decide wisely which parts to skip - concentrate on what's most interesting to you.

1. Send me email at eb@cs.umb.edu from the email account you want me to use to communicate with you this semester.

2. Flesh out the mathematical autobiography you started in class. How have study and teaching mixed? If you wish you can speculate on how your mathematical autobiography might continue. In particular, you may want to discuss your fondest hopes and worst fears for this course.

3. What's your favorite mathematical theorem/idea/example/structure? (Describe it in as much or as little detail as you like.)

4. Read the Preface and the Introduction to the Sethuraman text. Write a few paragraphs with your first reactions to the style and content.

5. Read quickly through (but do not necessarily study) Chapter 1. In what sense is the material here new to you? If you have Courant and Robbins, read about the same material there (pp 43-48) and compare the two treatments. (In fact there's yet another proof of the Fundamental Theorem of Arithmetic in Courant on page 23 which I didn't know was there until preparing this assignment.)

6. Try Exercise 6 on page 24. (You may find this easy or impossible - we need to find out which.)

   Now that you know (because you proved it, or because you believe Sethuraman) that gcd(a_n, a_n-1) = 1 you know (From Theorem 1.6) that it is possible to find integers x and y such that

   	x an + y an-1 = 1
   

   So do it.

   Start small: fill in the table

   	n	an	an-1	x	y
   	3	2	1	 1	-1
   	4	3	2	 1	-1
   	5	5	3	-1	 2
   	6	8	5	 2	-3
   	7 
   	...
   

   and look for a pattern you can prove.

7. How fast is the Euclidean algorithm? Since the remainder at each step is less than the divisor, the Euclidean algorithm finds gcd(a,b) in at most min(a,b) steps. But in fact it's much faster.

   See if you can show that two steps reduces one of the number to at most half its size. Conclude that the algorithm takes on the order of log₂(a) steps - that's much less than a.

   Google properly queried will probably find various analyses of the speed of the Euclidean algorithm. If you can't solve the problem yourself (or even if you can) you might want to look some up, and report on what you find. This kind of research is acceptable (unless I explicitly specify otherwise) under two conditions:

   • You must understand what you discover, and describe it in such a way that you demonstrate your understanding
   • You must acknowledge the sources you used.