Suppose is a solution to in . Show that . Deduce that if we fix any of , , or , then there are only finitely Pythagorean triples .
Suppose . Now , and . (We may assume that .) Now . Since and are integers, .
With fixed, there are only finitely many possible , as they must satisfy . With and fixed, is determined, so there are only finitely many solutions with fixed . Symmetrically, there are finitely many solutions with fixed .
With fixed, we have , so there are certainly only finitely many solutions.