Hi! Can anyone provide a proof for the following:
The following formula
generates all the positive integers that satisfy .
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. .