Dickson's conjecture on simultaneous prime values of linear polynomials

Let $k\in\boldsymbol{N}_{\geq 1}$ and let $bar f=(f_i:i<k)$, where $f_i(x)=a_i x+b_i$ with $a_i,b_i$ non-negative integers and $a_igeq 1$. Assume that no integer $ngeq 1$ divides all products $\prod_{i<k}f_i(s)$ simultaneously for every non-negative integer $s$. Dickson's conjecture. There are infinitely many natural numbers $m$ such that $f_i(m)$ is prime for every $i<k$. This is a classical unresolved conjecture in prime-number theory, generalizing several simultaneous-prime-value problems.