Resources for Pythagorean Triples
An Internet search for "Pythagorean triples" will yield literally hundreds of sites that contain a variety of information on the subject. We suggest that you resist looking at them until you run out of your own ideas. If you can't resist, here are a few good sites on the subject.
Eric Rowland's page
Fred Curtiss primitive Pythagorean triples page
William Richardson's page
One particularly interesting book, and probably the oldest one on the subject, is Leonardo Pisano Fibonacci's Liber quadratorum. Yes, the same guy who brought us the sequence 1, 1, 2, 3, 5, 8, 13, . . . wrote a book on Pythagorean triples. The hint about the sum of odd integers in the Hints section is the first method that he used. The original Latin edition, written in 1225, was translated in 1987.
Leonardo Pisano Fibonacci, L. E. Sigler (Translator), The Book of Squares, Academic Press, 1987.
Here is an excellent set of notes on student research that touches on Pythagorean triples:
Stephen I. Brown, Student Generations (HiMap Module 7), COMAP, 1981.
This article expands on the ideas raised on this site:
Ernest J. Eckert, Primitive Pythagorean Triples, The College Mathematics Journal, 23 (1992) 413-417.