Bunyakovsky's conjecture on prime values of polynomials

Let

$$f(x)=\sum_{j=0}^{d}a_jx^j\in\mathbb{Z}[x]$$

be an irreducible polynomial with positive leading coefficient and satisfying $\mathrm{gcd}(a_1,\ldots,a_d)=1$.

Bunyakovsky's conjecture. The value $f(n)$ should be prime for infinitely many positive integers $n$.

This is a classical conjecture on prime values of irreducible polynomials and would, for example, imply the existence of infinitely many repunit primes with a fixed number of digits in the applications discussed in the paper. It remains open in general.