Dickson's conjecture

Statement

Suppose $a_1,a_2,\dots ,a_k,b_1,b_2,\dots ,b_k$ are integers with all the $a_i\ge 1$. 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