My book on number theory says that Radius of Circumcircle formed by triangle made of Pythagorean triplet cannot be an Integer which seems wrong to me.
Considering a triangle with sides 6,8,10, the radius would obviously be a 5 which is a integer. Am I missing something or is the book wrong?
It says since X is a even number. Why does X have to be Even? Is it just a copyright trap?
Perhaps they mean primitive Pythagorean triples. Those do have to have odd hypotenuses. All primitive Pythagorean triples (and some non-primitive ones) can be generated via the pattern $\{u^2-v^2, 2uv, u^2+v^2\}$, where $u$ and $v$ are positive integers, and $u^2+v^2$ is the hypotenuse.
Note that the leg $2uv$ is clearly even. Now suppose the hypotenuse $u^2+v^2$ were also even; then $u^2-v^2$ would have to be even as well. But then all three sides would be even, and the triple would not be primitive.
Their example is just that—an example. Not all primitive triples are of the form they give. For instance, $5$-$12$-$13$ clearly isn't.