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Math Help - Phi function proof

  1. #1
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    Phi function proof

    Let & denote the phi function.

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

    Thanks for the help...
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by jzellt View Post
    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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