This assignment is due next Thursday (9/24) but start it over the weekend so we can discuss it some on Tuesday.

1. Explore as thoroughly as you can the solutions (if any) to each of the Diophantine equations

   	x2 + 2y2 = n
   	x2 - 2y2 = n
   

   In particular, see if you can make a conjecture about which prime values of n allow a solution. Then see how much of that conjecture you can prove (maybe not much)!

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

   	x2 - 2y2 = +-1
   

   That's looking for a square that's close to twice another square. (We know from the first class that you can never find a square that's exactly twice another square.) When you find a solution, take a look at x/y, which will be a rational number whose square is close to 2. How close (in terms of y)?

   You should be able to find a few small solutions by inspection and then find a pattern that allows you to prove that there are infinitely many solutions. It's harder to show that the infinite list you've found includes all the solutions.

   You might enjoy checking out your solutions in the On Line Encyclopedia of Integer Sequences: www.research.att.com/~njas/sequences/index.html

3. Solve the Diophantine equation

   	377x + 610y = 1
   

   or prove it has no solutions.

4. Your solution to the previous problem should suggest some theorems about Fibonacci numbers. Make some conjectures. Prove them if you can.

5. We've now looked at the expressions

   	x2 +- y2
   

   and

   	x2 +- 2y2
   

   to see which integers they represent (and how often). What problem or problems come next?