# 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
4 years, 2 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

$$n|\varphi(p^n-1)$$

- 4 years, 2 months ago

Beautiful. :D

- 4 years, 2 months ago