Integer solutions to $x^2 + (x-y)^2 = 17^2$, with $x\ge0$

Question

I need help finding integer solutions to $$x^2 + (x-y)^2 = 17^2$$ with $x\ge0$

I know that this is of the form of the pythagorean theorem but I am quite unsure about how to proceed in finding all integer solutions for it.

This question was given to me by my teacher as a challenge question in our precalculus course.

Answer 1

Your question is solved once you find all integer solutions to $$a^2+b^2=17^2$$ This equation of course admits a solution as $17^2=17\times 17$ and hence the only primes dividing it are of the form $4k+1$ (if this statement doesn't make sense, you should check out Fermat's Two Square Theorem or just check that one solution is $(15,8)$).

Once you have one solution, you can find all the solutions using this method.