How would you solve diophatine equations of the form $x^2+y^2=25$? I know how to solve linear diophatine equations but I have not done any of quadratic form before. I could use trial and error because the numbers are small, but is there a more general method that can be used?
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Observe it suffices you look for solutions with $|y|\leqslant 5$, $|x|\leqslant 5$, and in fact any solution $(x_0,y_0)$ gives rise rise other solutions by changing signs. So any solution is paired up with other three solutions. Note if a coordinate is zero, you get repeated solutions. The number of integers points with nonzero coordinates and $|y|\leqslant 5$, $|x|\leqslant 5$ is $25$, so there aren't many options. Also, use some parity arugments. – Pedro Apr 17 '14 at 18:05
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But you still need trial and error to find those solutions? Is there another method? – 1110101001 Apr 17 '14 at 18:06
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See also: http://math.stackexchange.com/questions/753294/find-two-triangles-of-longest-side-length-25/ – Martin Sleziak Apr 17 '14 at 18:49
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All primitive pythagorean triples are of the form $r^2+s^2, r^2-s^2, 2rs$. Now just do case work on $6gcd(x,y)$ to get that they are coprime. Then its pretty much guess and check.
Sandeep Silwal
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