Prime values of irreducible higher-degree polynomials

Determine whether every irreducible polynomial P(x)∈Z[x] of degree at least two with no fixed divisor takes prime values for infinitely many nonnegative integers.

Background

The wisde-prime problem is an instance of the broader problem of determining whether polynomial sequences contain infinitely many primes. The paper notes that the relevant nine cubic polynomials are irreducible and have no fixed divisor after the stated filtering.

Even this general prime-values problem is unresolved for irreducible polynomials of degree at least two. The single-polynomial Bateman–Horn conjecture gives a conjectural asymptotic, but no unconditional proof is known.

References

Even for an irreducible polynomial $P(x)\inZ[x]$ of degree at least two with no fixed divisor, it remains unknown whether $P(n)$ is prime for infinitely many $n\inZ_{\ge0}$.

— Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms  (2609.03839 - Swanson, 3 Sep 2026) in Section 5, immediately after Corollary 5.3

Let $P(x)\inZ[x]$ be irreducible with positive leading coefficient and no fixed divisor. Then\footnote{Here $f(N)\sim g(N)$ means that $f(N)/g(N)\to1$ as $N\to\infty$.} $#{1\le n\le N:P(n)\text{ is prime}}\sim C(P)\frac{N}{\deg(P)\log N},$ where

— Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms  (2609.03839 - Swanson, 3 Sep 2026) in Conjecture 5.4, Section 5