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When I was eating lunch, I made up the following problem.

Does $n|\phi(m^n-1)$ where $m,n\in\mathbb{Z}^+$ and $m>1$? Prove or give a counterexample.

I asked my friend Issac Yiu to help me as I don't know how to solve, but we still don't have a solution.

Attempt:

When $n=1$, $1|m-1$

When $n=2$, $m^2-1>2,$ so $2|\phi(m^2-1)$

Me and Issac can both do the case $n=1$ and $2$, but not others.

Any help is appreciated!

Click here to go to a discussion in Brilliant about the problem. (The discussion is made by Issac Yiu.)

Edit: Can anyone not use group theory to solve this problem? The other questions which asks this problem were solved by group theory or I don't understand.

Culver Kwan
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  • Your updated question has also been answered: https://math.stackexchange.com/questions/594233/prove-that-n-divides-phian-1-where-a-n-are-positive-integer-without – darij grinberg Aug 08 '19 at 10:16

1 Answers1

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Mod $m^n-1$, $m$ has multiplicative order $n$. By Lagrange the conclusion follows.

Aphelli
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