Find element of specific order in group of a large prime power

Question

Consider the group $\mathbb{Z}^{\ast}_p$ where

$p = 8631907183416457710036142986950248939902071925468562052182263723074665 2812723253836453003270766004614328295206393288529828311628415785652726 6539861069097142611570511777538875474993819960760889160802939$

I know that $(p-1)/2$ is a prime number as well. Is there an element in the group of order $(p-1)/2$ and, if so, how can I see this?

Answer 1

There is such element in the group. If $p$ is prime, then $\Bbb Z_p^*$ has a primitive element, that is, an element of order $p-1$. Let $a$ be this element.

Then $a^2$ has order $(p-1)/2$.