Is L2 = {a^n| n is a product of one or more primes} regular? I am having a hard time proving the following with pumping lemma:
Is $L_2 = \{a^n \mid \text{$n$ is a product of one or more primes}\}$ regular?

Here's what I have so far:


Suppose $L$ is regular, and let $m$ be the constant pumping lemma.
Let then $w = a^m$ where $m$ is a product of at least one prime number, therefore $w$ is in the language $L_2$ and $|w| \geq m$.
Then $w = xyz$, where $|xy| \leq m$ and $|y| \geq 1$.
Then $y = a^j$ where $j \geq 1$.


This is where I'm not sure what to do:
Pumping down we get $w_2 = a^{m-j}$

I don't know where to keep going from here since I don't even know if pumping down is the right thing to do.
According to the Fundamental Theorem of Arithmetic, any integer $&gt;1$ can be written as a product of one or more primes (in a unique way).  So, it seems that your language can be simplified as $\{a^n\mid n\geq 2\}$.