Date: Mon, 8 Dec 1997 22:51:17 -0500

From: Alex Bogomolny

Dear Shameq:

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 = 2x^{2}+2x, c = b+1 you automatically have a formula for the perimeter: p = 2x^{2}+6x+3 and another for the area A = (2x+1)(x^{2}+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} + (2x^{2}+2x)^{2} = (2x^{2}+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 = x^{2} - 1, c = b+2.

Best regards

Alexander Bogomolny