# How many primes of $ap+b$ with $(a,b)=1$ and $p$ be any primes?

There is a famous Dirichlet's result, which says: There are infinitely many primes of form $an+b$, where $(a,b)=1$ and $n\in \mathbb{N}$.

Denote the set of all primes by $\mathbb{P}$. Given two coprime numbers $a$, $b$, how many primes of form $ap+b$, where $p\in\mathbb{P}$? For example, how many primes of form $2p+1$, where $p\in\mathbb{P}$?

More generally, for a given irreducible primitive polynomial $f(x)$, is $\{f(p)~|~p\in\mathbb{P}\}\cap\mathbb{P}$ a finite set? Did somebody study this problem?

I also need to cite this Dirichlet's result. Could anyone tell me from which book or article I can cite this theorem?

• $a$ and $b$ are fixed integers and you're iterating over the prime numbers for $p$, right? – Stefan4024 Mar 1 '17 at 8:04
• Yes. @Stefan4024 – user44312 Mar 1 '17 at 8:05
• You could take a look at: en.wikipedia.org/wiki/Sophie_Germain_prime – Mastrem Mar 1 '17 at 12:31
• Thanks. It's very helpful. @Mastrem – user44312 Mar 4 '17 at 8:36

No univariate polynomial with integer coefficients and degree greater than $1$ is known to produce infinite many primes.
But many such polynomials , as $x^2+1$ are conjectured to produce infinite many primes.