Infinitely many pairs of primes?

Question

I want to know how many (infinitely many) pairs of primes are known.

For convenience, let me give two definitions.

For any nonconstant polynomial $f(x)\in \mathbb{Z}[x]$, define $A_f=\lbrace f(p) \in \mathbb{Z} \mid \text{Both of $p$, $f(p)$ are primes}\rbrace$.

Also, define $P=\lbrace f(x)\in \mathbb{Z}[x] \mid \lvert A_f\rvert=\infty\rbrace$, where $\lvert A\rvert$ is the cardinality of set $A$.

Let me give some examples. If $f(x)=x$, then it is (trivial) prime pairs (i.e., $f(x)=x \in P$).

If $f(x)=x+2$, then the case is that the famous twin prime conjecture (i.e., twin prime conjecture is equivalent to determine that $f(x)=x+2$ is in $P$ or not).

I also heard that the case of $f(x)=4x+1$ is also (famous) conjecture.

My question is whether there are any nontrivial polynomials which are known to lie in $P$?

Answer 1

As a special case of Schinzel's hypothesis H (a well-known open problem) if $f(x)$ is irreducible, has positive leading coefficient and has no “fixed divisor” then $f(x)$ should lie in what you call $P$. But nothing of this sort has been proved and I am confident that not a single polynomial (other than $x$) has been proved to be in what you call $P$. For polynomials of degree at least two, it's even worse, there is no polynomial which has been proved to take prime values at infinitely many integers (let alone primes).