Let $\displaystyle k $ be positive integer $\displaystyle , n=3^k+2$ composite, squarefree
Show that $\displaystyle \Phi(n) \not= 2(3^{k-1}+1)$
For any natural integer $\displaystyle m=p_i^{a_i}\cdot\ldots\cdot p_k^{a_k}$ , with $\displaystyle p_i$ primes, $\displaystyle 0<a_i\in\mathbb{N}$, we have that $\displaystyle \phi(m)=m\prod\limits_{1\le i\le k}\left(1-\frac{1}{p_i}\right)$, so if $\displaystyle 2\mid \phi(n)$, then n has to be divisible by a power of 2 greater than 1, and since $\displaystyle n=3^k+2$ then n is odd...
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