If you let , then k must be an integer and x must be an integer solution to . Hence, the discriminant of this polynomial, which is , must be a square. Finally, let . Then must be a square. Now use your knowledge of pythagorean triples and multiples of them to find solutions:
Now we need hypotenuses . From these triples we get the solutions .
This yields the following solutions for k:
That gives you seven quadratic equations to solve for x, some of which may not yield integer solutions.