Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

For p a prime and n a positive integer, consider the group of units, $(\mathbb{Z}/(p^{n}-1)\mathbb{Z})^{\times}$. How can I go about to find the order of $\bar{p}$?

share|improve this question
    
@Zev, $p^{n}=1$, but how do I know this is the order of p? –  Edison Sep 28 '11 at 1:58
    
I've made my comment into an answer. –  Zev Chonoles Sep 28 '11 at 2:06
add comment

1 Answer 1

up vote 3 down vote accepted

Clearly, $p^n\equiv 1\bmod (p^n-1)$, so the order of $p$ is $\leq n$. But if $p^k\equiv 1\bmod (p^n-1)$, then $$(p^n-1)\mid (p^k-1)$$ and if $k<n$ then $p^k-1<p^n-1$, so this is impossible. Therefore the order must be precisely $n$.

share|improve this answer
1  
Note this has nothing to do with p being a prime. –  KCd Sep 28 '11 at 3:19
    
@Zev, so when you write $p^{k}= 1 mod (p^{n}-1)$ you are assuming that $k>n$, but in your conclusion you assume $k<n$? I think I am misunderstanding something. –  Edison Sep 28 '11 at 3:53
    
@El G: I demonstrate that $p^k\equiv 1\bmod (p^n-1)\implies k\geq n$, using a proof by contradiction (if we had $k<n$, we would get the contradiction that $p^n-1\leq p^k-1$ and $p^k-1<p^n-1$). –  Zev Chonoles Sep 28 '11 at 4:01
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.