Are there any short, elegant proofs known for the identity $\varphi(p^{k})=p^{k}-p^{k-1}$ ? (Here $\varphi$ is Euler's totient function and $p$ is a prime.) The standard combinatorial proof goes like this: In the set $\left\{ 1,2\ldots,p^{k}\right\} $ there in total $p^{k}$ number. Split ...