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 = 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.

Best regards
Alexander Bogomolny

Copyright © 1996-2008 Alexander Bogomolny

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