Dickson's conjecture

In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by Dickson (1904) that for a finite set of linear forms $a 1 + b 1 n$ , $a 2 + b 2 n$ , ..., $a k + b k n$ with $b i \geq 1$ , there are infinitely many positive integers $n$ for which they are all prime, unless there is a congruence condition preventing this (Ribenboim 1996, 6.I). The case k = 1 is Dirichlet's theorem.

Two other special cases are well-known conjectures: there are infinitely many twin primes (n and 2 + n are primes), and there are infinitely many Sophie Germain primes (n and 1 + 2n are primes).

Generalized Dickson's conjecture

Given n polynomials with positive degrees and integer coefficients (n can be any natural number) that each satisfy all three conditions in the Bunyakovsky conjecture, and for any prime p there is an integer x such that the values of all n polynomials at x are not divisible by p, then there are infinitely many positive integers x such that all values of these n polynomials at x are prime. For example, if the conjecture is true then there are infinitely many positive integers x such that $x^{2}+1$ , $3x-1$ , and $x^{2}+x+41$ are all prime. When all the polynomials have degree 1, this is the original Dickson's conjecture. This generalization is equivalent to the generalized Bunyakovsky conjecture and Schinzel's hypothesis H.