Dickson's conjecture

In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by that for a finite set of linear forms $a_{1} + b_{1}n$, $a_{2} + b_{2}n$, ..., $a_{k} + b_{k}n$ with $b_{i} ≥ 1$, there are infinitely many positive integers $n$ for which they are all prime, unless there is a congruence condition preventing this. 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).

Dickson's conjecture is further extended by Schinzel's hypothesis H.

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 more general conjecture is the same as the Generalized Bunyakovsky conjecture.