Schinzel’s Hypothesis H

Let a set of irreducible polynomials $P_1,P_2,P_3,\mathrm{\dots },P_k$ with integer coefficients have the property that for any prime $p$ there exists some $n$ such that $P_1(n)P_2(n)\mathrm{\dots }{P}_k(n)\not\equiv 0(modp)$. Schinzel’s Hypothesis H that there are infinitely many values of $n$ for which $P_1(n),P_2(n),\mathrm{\dots },$ and $P_k(n)$ are all prime numbers.

The 1st condition is necessary since if $P_i$ is reducible then $P_i(n)$ cannot be prime except in the finite number of cases where all but one of its factors are equal to 1 or -1. The second condition is necessary as otherwise there will always be at least 1 of the $P_i(n)$ divisible by $p$; and thus not all of the $P_i(n)$ are prime except in the finite number of cases where one of the $P_i(n)$ is equal to $p$.

It includes several other conjectures, such as the twin prime conjecture.

Title	Schinzel’s Hypothesis H
Canonical name	SchinzelsHypothesisH
Date of creation	2013-03-22 15:11:43
Last modified on	2013-03-22 15:11:43
Owner	jtolliver (9126)
Last modified by	jtolliver (9126)
Numerical id	5
Author	jtolliver (9126)
Entry type	Conjecture
Classification	msc 11N32