Dickson's conjecture: Difference between revisions

Latest revision as of 21:33, 29 January 2014

Suppose $a_{1}, a_{2}, \dots, a_{k}, b_{1}, b_{2}, \dots, b_{k}$ are integers with all the $a_{i} \geq 1$ . Then, consider the polynomials:

$f_{i} (x) : = a_{i} x + b_{i}, i \in {1, 2, \dots, n}$

Then, one of the following is true:

There is a prime number $p$ such that the product $\prod_{i = 1}^{k} f_{i} (x)$ is $p$ times an integer-valued polynomial. In other words, one of the polynomials $f_{i} (x)$ is always congruent to 1 modulo $p$ .
There exist infinitely many [[natural number]s $n$ for which all the values $f_{i} (n)$ are simultaneously prime.

Schinzel's hypothesis H generalizes from linear polynomials to polynomial of arbitrary degree.
Bateman-Horn conjecture further generalies Schinzel's hypothesis H by providing an asymptotic quantitative estimate of the frequency of occurrence of primes.

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 * There is a [[prime number]] <math>p</math> such that the product <math>\prod_{i=1}^k f_i(x)</math> is <math>p</math> times an integer-valued polynomial. In other words, one of the polynomials <math>f_i(x)</math> is always congruent to 1 modulo <math>p</math>.
 * There exist infinitely many [[natural number]s <math>n</math> for which ''all'' the values <math>f_i(n)</math> are ''simultaneously'' [[prime number|prime]].
+==Related facts and conjectures==
+===Stronger facts and conjectures===
+* [[Schinzel's hypothesis H]] generalizes from linear polynomials to polynomial of arbitrary degree.
+* [[Bateman-Horn conjecture]] further generalies Schinzel's hypothesis H by providing an asymptotic quantitative estimate of the frequency of occurrence of primes.
+===Weaker facts and conjectures===
+* [[Green-Tao theorem]]
+* [[Twin prime conjecture]]
+* [[Polignac's conjecture]]