Let & denote the phi function.

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

Thanks for the help...

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- Apr 8th 2010, 09:47 PMjzelltPhi function proof
Let & denote the phi function.

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

Thanks for the help... - Apr 8th 2010, 10:03 PMchiph588@
Assume $\displaystyle n $ is odd, $\displaystyle \phi(n) = \prod_{p_i} p_i^{\alpha_i-1}(p_i-1) $

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

$\displaystyle 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 $\displaystyle n $ is even. (The result follows similarly.)