Dickson's conjecture

Statement

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.

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