Let $\lambda$ denote the Carmichael function and $\varphi$ Euler's totient function. Furthermore, let $p$ denote any prime number and $k\in\mathbb{N}$. The wikipedia article about $\lambda$ states: $$\lambda(p^k)=\begin{cases}\frac{1}{2}\varphi (p^k)&\text{, if }p=2\wedge k>2\\\varphi (p^k)&\text{, else}\end{cases}$$
I want to prove this statement. So, let $a\in\mathbb{Z}$ with $\gcd(a,p^k)=1$. From Euler's theorem, it follows: $$a^{\varphi (p^k)}\equiv 1\mod p^k$$ Thus, we've got $$\lambda (p^k)\le \varphi (p^k)$$How do I need to proceed from here?
Best Answer
The key topic you want to look up is "primitive roots". Saying that $\lambda(p^k)=\phi(p^k)$ is equivalent to saying that $(\Bbb Z/p^k\Bbb Z)^\times$ is a cyclic group; this is what the known results on primitive roots assert. Any place that discusses primitive roots (for example, the book by Niven/Zuckerman/Montgomery) will also indicate how the situation is slightly different for powers of $2$, which will give you the special case as well.