Schinzel's hypothesis H
From Number
Contents
Statement
Suppose are all irreducible polynomials with integer coefficients and with positive leading coefficient, such that the product 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 satisfying the condition that 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 and . Also, the dependence is via Dickson's conjecture. |
Polignac's conjecture | open | We are dealing with the irreducible polynomails and where 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. |