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$

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$

Or simpler: $$ n(x)\ > \ \sqrt{P(x)}+1$$