# If a case of the Bunyakovsky conjecture holds for degree-$n$ $p(x)$, does it hold for all degree-$n$ $f(x)$ satisfying the criterion?

The Bunyakovsky conjecture states that if a polynomial $$f(x)$$ satisfies:

1. The leading coefficient is positive
2. The polynomial is irreducible over $$\mathbb{Z}$$
3. $$f(1), f(2) \dots$$ share no common factor (the coefficients of $$f(x)$$ should be relatively prime)

then the sequence $$f(1), f(2) \dots$$ contains infinitely many primes.

My question: if Bunyakovsky's conjecture does hold for a polynomial $$p(x)$$ of degree $$n>1$$ satisfying the above conditions, does that tell us anything about other polynomials of degree $$n$$ satisfying the above conditions? That is, if the conjecture holds true for a degree-$$n$$ polynomial, must it hold true for all other sufficient degree-$$n$$ polynomials? (I have no substantial reason to think it might, but I also have no substantial reason to think it might not, and it doesn't seem that far-fetched.)

If it holds for $$f(x)$$, then it also holds for $$f(x+k)$$ for any integer $$k$$. If $$f(x/m)$$ also has integer coefficients, then it also holds for $$f(x/m)$$. But I see no obvious reason (except the truth of the conjecture) why it should hold for arbitrary polynomials of the same degree satisfying the conditions.