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?
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?
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.