Euler Phi funtion

**wauwau** · December 23rd 2009, 03:13 AM

Let $k$ be positive integer $, n=3^k+2$ composite, squarefree

Show that $\Phi(n) \not= 2(3^{k-1}+1)$

**tonio** · December 23rd 2009, 05:40 AM

Originally Posted by wauwau

Let $k$ be positive integer $, n=3^k+2$ composite, squarefree

Show that $\Phi(n) \not= 2(3^{k-1}+1)$

For any natural integer $m=p_i^{a_i}\cdot\ldots\cdot p_k^{a_k}$ , with $p_i$ primes, $0<a_i\in\mathbb{N}$ , we have that $\phi(m)=m\prod\limits_{1\le i\le k}\left(1-\frac{1}{p_i}\right)$ , so if $2\mid \phi(n)$ , then n has to be divisible by a power of 2 greater than 1, and since $n=3^k+2$ then n is odd...

Tonio

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