Schinzel’s Hypothesis H

Let a set of irreducible polynomials P1,P2,P3,,Pk with integer coefficients have the property that for any prime p there exists some n such that P1(n)P2(n)Pk(n)0(modp). Schinzel’s HypothesisMathworldPlanetmathPlanetmath H that there are infinitely many values of n for which P1(n),P2(n),, and Pk(n) are all prime numbersMathworldPlanetmath.

The 1st condition is necessary since if Pi is reducible then Pi(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 Pi(n) divisible by p; and thus not all of the Pi(n) are prime except in the finite number of cases where one of the Pi(n) is equal to p.

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

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