a2 + b2 = c2

1. Show that in any ring the formulas

   	a = (u2 - v2)r
   	b = 2uvr					(PT)
   	c = (u2 + v2)r
   

   produce a Pythagorean triple (a,b,c) for any choice of ring elements u, v and r.

2. Prove that in the positive integers the Pythagorean triple is primitive just when r=1, u > v, and u and v are relatively prime and not both odd. (To prove this you might first want to see what happens when you systematically try examples in which each of the conditions fails. But remember that no examples can prove the theorem - they can just suggest a way to prove it.)

3. Show that every odd number can appear as a in a primitive Pythagorean triple (a,b,c) . Hint 1: consider (3,4,5), (5,12,13) and (7,24,25) and look for a pattern. Hint 2: Compute (n+1)² - n² and study (PT).

4. Try to prove that the formulas (PT) generate all the Pythagorean triples in the integers. This may well be too hard for you. Here's a way for you to learn something from this problem anyway. Find a proof somewhere (on the internet, in a book, perhaps in Courant and Robbins), read it, understand it and then rewrite it in your own words. Your rewrite should be a real one, not just a word for word paraphrase. You need to add stuff that convinces me that you really understand what you have read. Submit both your rewrite and a copy (xerox or otherwise) of the original.

5. How many Pythagorean triples are there in Z_n? This is a question I've never seen asked before. Although it's likely that someone has asked and answered it, I don't know the answer. I don't know whether the problem is easy or hard. I don't know whether the answer is interesting. Let's do some research.

   When n=2 there are three solutions: (0,0,0), (1,0,1) and (0,1,0). These three will work in any ring. So will (0,x,x) and (x,0,x), so we probably should not count any of the triples for which a=0 or b=0.

   In Z₃ there seem to be a few more: (2,0,1) is one of them. But that's really "the same" as (1,0,1) since in Z₃ 2 = -1 and we might not want to count (a,b,c) and (-a,b,c) as different Pythagorean triples (but then again we might want to count them as different if that helps a pattern emerge). When counting that way there are no more new triples in Z₃.

   To carry out more experiments we should be more systematic: list all the squares in Z_n, add each pair and see when squares result. For example, for Z₇ the squares are

   	02 = 0
   	12 = (-1)2 = 1
   	22 = (-2)2 = 4
   	32 = (-3)2 = 2
   

   so the Pythagorean triples are

   	(1,1,3)
   	(2,2,1)
   	(3,3,2)
   

   What next? Compute many more examples: certainly n=4,5,6, then maybe 11 and 13, in hopes that the primes may be more informative. But I don't know. You might want to work collectively as a class, with different people working out different examples so that together you have more data than any one of you could find on his or her own.