Let $p_m$ be the $m$th positive prime number in $\Bbb{Z}$. Then $f \in \Bbb{Z}[X]$ is irreducible if:
$$ \liminf\limits_{m \to \infty} \dfrac{\# \{f(n) \text{ is prime } : n \lt p_m \}}{m} \gt 0 $$
I think I have an elementary proof for it.
Proof: If the polynomial $f$ is reducible, then $f(n) = a(n)b(n)$ and so $f(n)$ can only be prime a finite number of times, no matter how far out $m$ goes. Insert argument about how a polynomial (namely $a$ or $b$) cannot infinitely take on values $\pm 1$, maybe talking about Taylor series for $\cos$ since the approximating polynomials "try" to do this.
Under an asymptotic density limit, if the top is finite, then clearly the $\lim \to 0$. $\square$
What do you think? How would you prove the converse (probably way harder)? Have I solved twin primes? J/k :)
EDIT:
Changed $\lim $ to $\liminf$ since who knows how the graph of the counting function behaves.