Dickson's conjecture

In number theory, Dickson's conjecture is the statement that for a finite set of linear forms $a_{1}+b_{1}n,a_{2}+b_{2}n,\dots,a_{k}+b_{k}n$ with each ⁠ $b_{i}\geq 1$ ⁠, there are infinitely many positive integers $n$ for which they are all prime, unless there is a congruence condition preventing this. The conjecture is named after Leonard Dickson, who first proposed it in 1904. The case $k=1$ is Dirichlet's theorem. Two other special cases are well-known conjectures: that there are infinitely many twin primes (⁠ $n$ ⁠ and $n+2$ are primes), and that there are infinitely many Sophie Germain primes (⁠ $n$ ⁠ and $2n+1$ are primes).