Show that for $p \neq 2$ not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square. Show that for $p\neq2$, not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square of an element in $\mathbb{Z}/p\mathbb{Z}$. (Hint: $1^2=(p-1)^2=1$. Deduce the desired conclusion by counting).
So far I have that $1=p^2-2p-1\Rightarrow p^2-2p=0\Rightarrow p^2=2p$, but I don't know where to go from here. I also don't fully understand what it means to deduce it by counting.

It means you should count how many squares there are, they are certainly less than $p$ since $1^2=-1^2$ (and $1\neq -1$ when $p\neq 2$)
