The Bunyakovsky conjecture states the following :
Let $f$ be an irreducible polynomial and $d$ denote the gcd of the set $f(a)$, where $a$ runs over the integers. Then, $f(a)/d$ is prime for infinite many integers $a$.
I found statements about the converse:
If $a$ is large enough, then if $f(a)$ is prime, then $f$ is irreducible. This "large enough" is precised.
But the case $d>1$ is not considered!
There remain two possibilities :
There is a translation to a function $g$ with $d = 1$, such that $f$ is irreducible if and only if $g$ is irreducible.
If $a$ is large enough, then if $f(a)/d$ is prime, then $f$ is irreducible.
Which of the two possibilities can be used to check polynomials with $d > 1$? And if the second possibility works, which number is large enough ?