Schinzel's hypothesis H

Statement

Suppose $f_{1},f_{2},\dots ,f_{r}$ are all irreducible polynomials with integer coefficients and with positive leading coefficient, such that the product $\prod _{i=1}^{r}f_{i}$ does not have any fixed divisors, i.e., it cannot be expressed as a proper multiple of an integer-valued polynomial. Schinzel's hypothesis H states that there are infinitely many natural numbers $n$ satisfying the condition that $f_{1}(n),f_{2}(n),\dots ,f_{r}(n)$ are all simultaneously prime.

Related facts and conjectures

Stronger facts and conjectures

• Bateman-Horn conjecture provides a quantitative estimate of the frequency with which we get primes.

Weaker facts and conjectures

Fact or conjecture	Status	How it fits with Schinzel's hypothesis H
Dickson's conjecture	Open	We are dealing with linear polynomials.
Bunyakovsky conjecture	open	We are dealing with only one irreducible polynomial of degree two or higher
Dirichlet's theorem on primes in arithmetic progressions	proved	We are dealing with one irreducible polynomial of degree one
Twin prime conjecture	open	We are dealing with the irreducible polynomials $x$ and $x+2$. Also, the dependence is via Dickson's conjecture.
Polignac's conjecture	open	We are dealing with the irreducible polynomails $x$ and $x+2m$ where $m$ is a positive integer.
Green-Tao theorem	proved	The theorem states that the sequence of primes contains arithmetic progressions of arbitrary length. The Green-Tao theorem can be viewed as a corollary of Schinzel's hypothesis H if we view it as a collection of statements, one about the existence of arithmetic progressions of a specific length. Each such statement is substantially weaker than what we'd get from Schinzel's hypothesis H (which not only guarantees the existence of such arithmetic progressions, but also allows us to choose a common difference for the arithmetic progression that satisfies some divisibility conditions). The dependence is via Dickson's conjecture.