Does every mod p have at least one element with a non-identical inverse?
I very much suspect this is true, but how can I prove it? For example, in mod 5, some elements have inverses that are not themselves ${2,3}$ and some have themselves as inverses ${1,4}$. Am I assured that every prime $p\gt 2$ will have at least one element that is not its own inverse (almost certainly yes)? How do I prove that?
On
An element which is its own inverse modulo $p$ is represented by an integer $x$ such that $x^2\equiv 1\pmod{p}$, that is, $p\mid (x^2-1)$ which means $$ p\mid x-1 \quad\text{or}\quad p\mid x+1 $$ In other words, either $x\equiv 1\pmod{p}$ or $x\equiv p-1\pmod{p}$.
So, as soon as $p>3$, there are elements like you're looking for.
$x$ is its own inverse $\bmod p$ iff $x^2\equiv 1\pmod p$ and this has only the two solutions $\pm1\pmod p$ (put differently, $p\mid x^2-1=(x-1)(x+1)$ implies $p\mid x-1$ or $p\mid x+1$).