Schinzel's hypothesis H
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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. |
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. |