Schinzel's hypothesis H: Difference between revisions

Revision as of 21:19, 29 January 2014

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

Weaker facts

Fact or conjecture	Status	How it fits with Schinzel's hypothesis H
Bunyakovsky 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 prime conjecture	open	We are dealing with the irreducible polynomials $x$ and $x+2$

@@ Line 4: / Line 4: @@
 ==Related facts and conjectures==
+===Weaker facts===
 {| class="sortable" border="1"
@@ Line 12: / Line 14: @@
 | [[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>
+| [[Twin prime conjecture]] || open || We are dealing with the irreducible polynomials <math>x</math> and <math>x + 2</math>
 |}