Simply said, radius $r$ can go to $\infty$.

$(4,0)$ and $(-4,0)$ lie of the a circle, which means the circle is symmetric about y-axis

Let $(h,k)$ be the center of the circle.
Since the circle is symmetric about y-axis, it makes $h = 0$

Equation of this circle -> $(x-h)^2 + (y-k)^2 = r^2$

Now $(4,0)$ and $(-4,0)$ lie of the a circle.
$(4-0)^2 + (0-k)^2 = r^2$ and $(-4-0)^2 + (0-k)^2 = r^2$

$(4)^2 + (k)^2 = r^2$

$4^2 + k^2 = r^2$

$r$ can be interpreted as hypotenuse of a right-angles triangle, considering the Pythagorean Theorem into consideration.

But here is the catch, $r$ is not restricted to be an integer and hence its value can literally go to infinity.

Hence, Answer is E