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?



Conjecture on prime numbers
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.


