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Given a prime $p$, let $a_n=pn+n-1$.

I have noticed that $\forall{p}\exists{n}\in[2,p]:a_n\in\mathbb{P}$.

For example: $p=7,a_3=23,a_4=31,a_6=47$.

What is this conjecture called, and has it been proved?

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This is related to Linnik's theorem: http://en.wikipedia.org/wiki/Linnik%27s_theorem . See in particular the conjecture on this wikipedia page:

It is also conjectured that: $p(a,d) < d^2$,

where $p(a,d)$ is the least prime in the arithmetic progression $a + nd$.

Note that the fact that $p$ itself is prime is irrelevant.

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  • $\begingroup$ Thank you. This link states that $a\geq1$, whereas in my question, $a=-1$. How do you workaround this "conflict"? $\endgroup$ – barak manos Oct 9 '14 at 9:58
  • $\begingroup$ You could just take $a=d-1$. $\endgroup$ – Gerry Myerson Oct 9 '14 at 10:26

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