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.
