Finding Pythagorean Triple that sums to 1000
Any pythagorean triple is in form of


  $$ k(u^2 - v^2), 2kuv, k(u^2 + v^2) $$


where u, v, and k are positive integers with u > v, u − v odd, and with u and v  coprime. The sum is therefore


  $$ ku(u + 2v) $$


(the restrictions still apply).

Now your job is factor 1000 into 3 terms (not so many ways to do so, 1000 = 5*5*5*2*2*2) and determine if those terms can be represented with u and v being coprime with an odd difference.