Question

Pythagorean triples are given by the formulas x2 - y2, 2xy, and x2 + y2. Use the formulas for the Pythagorean triples to prove why it is not possible for a right triangle to have legs with lengths of 16 and an odd number.

Answer 1

Let $k$ be an integer. Suppose there is a triangle with legs of length 16 and $2k+1$. Then by the Pythagorean theorem, the length of the hypotenuse should be

$\sqrt{16^2+(2k+1)^2}=\sqrt{4k^2+4k+257}$

The formulas for Pythagorean triples say that if the legs are integers, then so must be the hypotenuse, because if $x=16$ and $y=2k+1$ are integers, then so are $x^2-y^2$, $2xy$, and $x^2+y^2$.

However, $4k^2+4k+257$ is not a perfect square trinomial, which means for any integer $k$, the length of the hypotenuse is not an integer, so such a triangle doesn't exist.