Let $n_k$ for $k=1,2,...,i$ be a finite sequence of positive integers, with $i>1$ and $n_1=0$. If there is a prime p such that for every positive integer m, one or more integers in {${(m+n_k)|1\leq k\leq i}$} is dividable by $p$, then $n_k$ is bad.
If $n_k$ is not bad, is there any disproof to the claim that there exist infinitely many primes j, such that all {${(j+n_k)|1\leq k\leq i}$} are prime?
(Clearly we cant prove it)
And two, how does one determine if an $n_k$ is bad? Can there be a finite covering set of primes for {${(m+n_k)|1\leq k\leq i}$} for all m, such that the same prime is not required for every m?