What does $(n|p)=1$ mean?

Question

My number theory book mentions the following condition:

$(n|p)=1$, where $p$ is prime.

What does $(n|p)=1$ mean? I used to think $n|p$ implies that $n$ divides $p$.

Answer 1

This is the Legendre symbol:

Definition. Let $ p $ be a prime number and $ n \in \Bbb{Z} $. Then $$ (n|p) \stackrel{\text{df}}{=} \begin{cases} 1 & \text{if $ n \equiv x^{2} ~ (\text{mod} ~ p) $ has a solution and $ n \not\equiv 0 ~ (\text{mod} ~ p) $}, \\ -1 & \text{if $ n \equiv x^{2} ~ (\text{mod} ~ p) $ has no solution}, \\ 0 & \text{if $ n \equiv 0 ~ (\text{mod} ~ p) $}. \end{cases} $$

As Jyrki has mentioned in his comment above, this is usually typeset as $ \left( \dfrac{n}{p} \right) $ instead.