Hi! Can anyone provide a proof for the following:

The following formula

generates all the positive integers that satisfy .

Thanks!

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- August 8th 2010, 08:12 PMTerenceCSAbout Pythagorean triple...
Hi! Can anyone provide a proof for the following:

The following formula

generates all the positive integers that satisfy .

Thanks! - August 8th 2010, 08:26 PMundefined
I believe we can use an argument similar to the one in this thread post #22.

- August 8th 2010, 10:15 PMBruno J.
What undefined suggests is good! Here is another way.

You can suppose that are relatively prime. Then it is impossible for both to be odd (because then , which is impossible), and impossible for both of them to be even, which would contradict the hypothesis that are relatively prime. Hence, say is even and are odd. Write . Now both are even, say ; moreover are relatively prime. Since is a square, both must be squares, so we have , i.e. .