A prime conjecture 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?
 A: I have cribbed this answer from some notes of Kiran Kedlaya. 
Let $H$ be a $k$-tuple of distinct integers. For prime $p$, let $\nu_H(p)$ be the number of residue classes modulo $p$ represented by elements of $H$. If there is a prime $p$ such that $\nu_H(p)=p$, then it's easy to see that there are only finitely many integers $n$ such that $n+h$ is prime for every $h$ in $H$. Hardy and Littlewood conjectured that if $\nu_H(p)\lt p$ for every prime $p$, then there are infinitely many integers $n$ such that $n+h$ is prime for every $h$ in $H$. They went farther and conjectured an asymptotic formula for the number of such $n$: the number of $n\le x$ such that $n+h$ is prime for every $h$ in $H$ is asymptotic to $${x\over(\log x)^k}\prod_p\left(1-{\nu_H(p)\over p}\right)\left(1-{1\over p}\right)^{-k}$$
I think it is safe to say that all number theorists believe the conjecture is true. This even though, at the present time, there isn't even a single example of a $k$-tuple $H$ with $k\ge2$ for which it has been proved that there are infinitely many $n$ such that $n+h$ is prime for every $h$ in $H$.
