5
$\begingroup$

How can I prove that the value of $\varphi(p^n-1)$ (where $p$ is prime and $n$ is some positive integer) is some multiple of $n$? The purpose of this is to prove that $n$ divides $\varphi(p^n-1)$.

$\endgroup$
1

1 Answer 1

6
$\begingroup$

Consider the group of units $\Bbb Z/D\Bbb Z^\times$ where $D=p^n-1$, $p$ a prime.

What is the order of $p$?

$\endgroup$
6
  • $\begingroup$ the order is n, I have proved that. Now I must prove that n divides ϕ(p^n-1), and I figured the best way to do so is to prove that phi(p^n-1) is a multiple of n, but I'm not exactly sure how... $\endgroup$ Nov 4, 2013 at 0:54
  • $\begingroup$ @MalcolmLazarow Don't you have a theorem that says that if $f$ is the order of $g\in G$, $G$ a group, then $g^k=1\iff f\mid k$? $\endgroup$
    – Pedro
    Nov 4, 2013 at 0:56
  • $\begingroup$ @PedroTamaroff that follows from lagrange right? $\endgroup$
    – Asinomás
    Nov 4, 2013 at 0:57
  • $\begingroup$ @Omnitic Not really. From the division algorithm and the definition of order. $\endgroup$
    – Pedro
    Nov 4, 2013 at 0:58
  • $\begingroup$ ah!! there we go! thanks so much! $\endgroup$ Nov 4, 2013 at 1:00

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .