Density of primes in a polynomial Consider that $p(x)$ is an irreducible polynomial with integer coeficients, that $\mathrm{gcd}$ of its coefficients is $1$. What is the natural density of the below set?
$$A = \{n\ |\ p(n)\ \text{is prime}\}$$
And can we say an statement like prime number theorem about $A$?
For example the density for $p(x)=x$ is zero and maybe in general case that is zero too.

we have two cases:


*

*$p(x)$ is a linear polynomial: in this case we have a prime number theorem in arithmetic progressions that will give the distribution of primes in them. look at this page. 

*degree of $p(x)$ is greater than 1: in this case approximatly the problem is open. for example we don't know that there is infinitly many prime of the form $n^2+1$ or not.
