Let [tex]k[/tex] be an integer. Suppose there is a triangle with legs of length 16 and [tex]2k+1[/tex]. Then by the Pythagorean theorem, the length of the hypotenuse should be
[tex]\sqrt{16^2+(2k+1)^2}=\sqrt{4k^2+4k+257}[/tex]
The formulas for Pythagorean triples say that if the legs are integers, then so must be the hypotenuse, because if [tex]x=16[/tex] and [tex]y=2k+1[/tex] are integers, then so are [tex]x^2-y^2[/tex], [tex]2xy[/tex], and [tex]x^2+y^2[/tex].
However, [tex]4k^2+4k+257[/tex] is not a perfect square trinomial, which means for any integer [tex]k[/tex], the length of the hypotenuse is not an integer, so such a triangle doesn't exist.
