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 $P_{1}(n)P_{2}(n)...P_{k}(n)\ot\\equiv 0(mod\\;p)$ . 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.

单词	SchinzelsHypothesisH
释义	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 $P_{1}(n)P_{2}(n)...P_{k}(n)\ot\\equiv 0(mod\\;p)$ . 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.
随便看	directional derivative DirectionalDerivativeDerivationOf directional derivative, derivation of DirectionCosines direction cosines DirectLimitOfAlgebraicSystems direct limit of algebraic systems DirectProductOfAlgebras direct product of algebras DirectProductOfModules direct product of modules DirectProductOfPartialAlgebras direct product of partial algebras DirectProductsInACategoryOfFuncoids Direct products in a category of funcoids DirectProductsOfGroups direct products of groups DirectProductsOfHomomorphisms direct products of homomorphisms DirectSum direct sum DirectSummand direct summand DirectSumOfBoundedOperatorsOnHilbertSpaces direct sum of bounded operators on Hilbert spaces