# 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 |