(which are relatively prime) must themselves be squares; let and . Both and must be odd since exactly one of or is even and the other is odd. Therefore, both and are even, and one of them is divisible by 4. Dividing them by two produces two more integers and , one of which is even by the previous sentence. Because is a square, and are the legs of another primitive Pythagorean triangle whose area is . Since is itself a square and since is even, is a square. Thus, any Pythagorean triangle with square area leads to a smaller Pythagorean triangle with square area, completing the proof. Notes References Arithmetic problems of plane geometry Diophantine equations Fermat's Last Theorem Squares in number theory Theorems in number theory