We say that a set $A\subseteq \mathbb{N}$ has *lower density 0* if $$\text{lim inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$


Given $A,B\subseteq \mathbb{N}$ we set $A\cdot B = \{a\cdot b: a\in A, b\in B\}$. The set $A^n$ for $n\in\mathbb{N}$ is defined inductively in the obvious manner.


Let $P$ be the set of prime numbers in $\mathbb{N}$. What is the smallest positive $m\_0\in\mathbb{N}$ such that the lower density of $P^{m\_0}$ is $>0$? And what is the lower density of $P^{m\_0}$?



Lower density of {primes} times themselves
There is no such $m\_0$, due to the [Erdős–Kac theorem](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem).


