The sides of a right triangle are all integers. Two of them are odd numbers that differ by . What is the smallest possible value for the third side?
Source: NCTM Mathematics Teacher, February 2006
We look at primitive triples of a right triangle because they are integers that have the smallest possible values. A property of primitive triples states that the length of one leg is odd, that of the other leg is even and that of the hypotenuse is odd.
Let represent the odd leg, the even leg, and the odd hypotenuse. By the Pythagorean theorem
must be a perfect square. The smallest perfect square greater than is
The smallest possible value for the third side equals .