Phi function proof

**jzellt** · April 8th 2010, 09:47 PM

Let & denote the phi function.

Show that there are no integers n with &(n) = 14.

Thanks for the help...

**chiph588@** · April 8th 2010, 10:03 PM

Originally Posted by jzellt

Let & denote the phi function.

Show that there are no integers n with &(n) = 14.

Thanks for the help...

Assume $n$ is odd, $\phi(n) = \prod_{p_i} p_i^{\alpha_i-1}(p_i-1)$

This means $n$ can only have one prime factor since all $p_i-1$ are even.

$n=p^{a+1}\implies p^a(p-1)=2\cdot7\implies p-1=14,\;p^a=1 \text{ or } p-1=2,\; p^a=7$ , but none of those options work.

I'll leave you to consider when $n$ is even. (The result follows similarly.)

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