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).