# Math Help - Number Theory

1. ## Number Theory

I want to determine the smallest prime $p$ such that
$

\left(\frac {2}{p}\right)=-1
$

and the order of $[2]_p$ is less than $p-1$.

Here $

\left(\frac {2}{p}\right)=-1
$

is Legendre symbol.

I have tried with different prime number but have not got any answer.

2. $p \equiv 3 \ \text{or} \ 5 \ (\text{mod } 8) \ \ \Rightarrow \ \ (2/p) = - 1$

With the help of Wiki's Table of Primitive Roots , the first prime that does not have 2 as its primitive root and is congruent to 3 or 5 is $p=13$

3. but the order of [2]_13 is 12, and I am looking for a p where order of 2 is less than p-1?

4. Originally Posted by peteryellow
but the order of [2]_13 is 12, and I am looking for a p where order of 2 is less than p-1?
If you look at the table o_O provided thou shall see that $13$ has primitive root $6$ and that $[2]_{13} = [6]_{13}^5$.
Thus, $[2]_{13}$ is not a primitive root.