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Divisibility of Euler's Totient Function

Let \(p\) be a prime number and \(n\) be a positive integer. Prove that \(\phi(p^n-1)\) is divisible by \(n\), where \(\phi\) denotes Euler's totient function.

Note by Finn Hulse
2 years, 8 months ago

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Firstly,

\(p^n\equiv1\mod(p^n-1)\) Note that n is the smallest number with that property, in other words n\(=ord_{p^n-1}(p)\)

Also,

\(p^{\varphi(p^n-1)}\equiv1\mod(p^n-1)\)

But the order must divide every number with that property , so

\(n|\varphi(p^n-1)\) Bogdan Simeonov · 2 years, 8 months ago

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@Bogdan Simeonov Beautiful. :D Finn Hulse · 2 years, 8 months ago

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