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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?

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I think you're looking for what's called "the prime $k$-tuples conjecture". – Gerry Myerson Jun 6 '12 at 22:48
Yes, this answered my question fully. – TROLLKILLER Jun 6 '12 at 22:57
@Gerry: Why not expand that into a full answer? Or Perhaps Deuce will answer his own question? – mixedmath Jun 7 '12 at 0:52
@mixedmath, didn't have time then, but I've done it now. – Gerry Myerson Jun 7 '12 at 5:47
up vote 3 down vote accepted

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 $H$.

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