# Conjecture on prime numbers

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?

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