Speculatively, as for now, we could assume that the smallest difference between $n(x)$ and  $\sqrt{P(x)}$ is for $x=17$ (see Stefan's comment). 
Thus, a natural improvement to your theorem (conjecture?) could be the following:
$$n(x)\geq P(x)^{a}$$
Where 
$a=\log_p716$ and $p=510510$, so
$ a\approx 0.50016$