What meaning could possibly $m\simeq_{prim}n$ have? For positive integers, what does $m\simeq_{prim}n$ means? I have this: Let $\alpha\in\mathbb Z \wedge n\;$ positive integer. If $\alpha\simeq_{prim}n$, then $\alpha^{-1}\equiv\alpha^{\varphi(n)-1}$(mod $n$). How could this make sense? Edit: $\varphi$ is Eulers totient function.

From the context, it seems this notations $\ a\simeq_{prim}b\ $ wants to mean that $a$ and $b$ are _coprimes_ , i.e. don't share any common prime divisor.

Then, the statement is simply another form of Euler's theorem, as we have $$1\equiv a^{\varphi(n)}=a\cdot a^{\varphi(n)-1} \pmod n,$$ so the multiplicative inverse of $a$ exists and is $a^{\varphi(n)-1}$ modulo $n$, whenever $a$ is coprime to $n$.