# Schinzel’s Hypothesis H

Let a set of irreducible polynomials $P_{1},P_{2},P_{3},...,P_{k}$ with integer coefficients have the property that for any prime $p$ there exists some $n$ such that $P_{1}(n)P_{2}(n)...P_{k}(n)\not\equiv 0(mod\;p)$. Schinzel’s Hypothesis H that there are infinitely many values of $n$ for which $P_{1}(n),P_{2}(n),...,$ 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 SchinzelsHypothesisH 2013-03-22 15:11:43 2013-03-22 15:11:43 jtolliver (9126) jtolliver (9126) 5 jtolliver (9126) Conjecture msc 11N32