Schinzel's hypothesis H: Difference between revisions

Revision as of 01:01, 3 July 2012

Statement

Suppose $f_{1},f_{2},\dots ,f_{r}$ are all irreducible polynomials with integer coefficients and with positive leading coefficient, such that the product $\prod _{i=1}^{r}f_{i}$ does not have any fixed divisors, i.e., it cannot be expressed as a proper multiple of an integer-valued polynomial. Schinzel's hypothesis H states that there are infinitely many natural numbers $n$ satisfying the condition that $f_{1}(n),f_{2}(n),\dots ,f_{r}(n)$ are all simultaneously prime.

Related facts and conjectures

Fact or conjecture	Status	How it fits with Schinzel's hypothesis H
Bunyakovksy conjecture	open	We are dealing with only one irreducible polynomial of degree two or higher
Dirichlet's theorem on primes in arithmetic progressions	proved	We are dealing with one irreducible polynomial of degree one
Twin primes conjecture	open	We are dealing with the irreducible polynomials $x$ and $x+2$

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-=Statement==
+==Statement==
 Suppose <math>f_1,f_2,\dots,f_r</math> are all irreducible polynomials with integer coefficients and with positive leading coefficient, such that the product <math>\prod_{i=1}^r f_i</math> does not have any ''fixed divisors'', i.e., it cannot be expressed as a proper multiple of an integer-valued polynomial. '''Schinzel's hypothesis H''' states that there are infinitely many natural numbers <math>n</math> satisfying the condition that <math>f_1(n), f_2(n), \dots, f_r(n)</math> are all ''simultaneously'' prime.
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 | [[Bunyakovksy conjecture]] || open ||We are dealing with only one irreducible polynomial of degree two or higher
 |-
 | [[Dirichlet's theorem on primes in arithmetic progressions]] || proved || We are dealing with one irreducible polynomial of degree one
 |-
 | [[Twin primes conjecture]] || open || We are dealing with the irreducible polynomials <math>x</math> and <math>x + 2</math>
 |}