Subject: Re: More on Pythagorean triples
Date: Mon, 8 Dec 1997 22:51:17 -0500
From: Alex Bogomolny
Your questions got me quite confused. Your formulas for the Pythagorean triples are quite wonderful. I am sure I haven't seen them before.
But once you have formulas for sides a = 2x+1, b = 2x2+2x, c = b+1 you automatically have a formula for the perimeter: p = 2x2+6x+3 and another for the area A = (2x+1)(x2+x).
The right question to ask is probably whether with your formulas you can get all possible Pythagorean triples. The answer is of course No: e.g., the triple (20,21,29) is not obtainable by your formulas.
As to whether one can have c = b+2, the only thing I may say is that your formulas are implied by (2x+1)2 + (2x2+2x)2 = (2x2+2x+1)2 so that, if you want to choose c = b+2, you'll have to modify the other two of your formulas.
For example, take a = 2x, b = x2 - 1, c = b+2.