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Math Help - euler phi-function

  1. #1
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    euler phi-function

    Hi,

    I need help with this proof:

    Prove that there are no integers n for which phi(n) = n/4

    Appreciate any assistance
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  2. #2
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    Quote Originally Posted by htata123 View Post
    Hi,

    I need help with this proof:

    Prove that there are no integers n for which phi(n) = n/4

    Appreciate any assistance
    Write n in a prime factorization and use formula for the phi-function.
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  3. #3
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    I did that but I'm still stuck. Any ideas on how I should proceed?
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  4. #4
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    Quote Originally Posted by htata123 View Post
    I did that but I'm still stuck. Any ideas on how I should proceed?
    Certainly n is divisible by 4 so we can write n=2^a\cdot m where a\geq 2 and m is a positive odd integer. Now, \phi(n) = \phi(2^a)\phi(m) = 2^{a-1}\phi(m) since (2^a,m)=1. While n/4 = 2^{a-2}m. Therefore, if \phi(n) = n/4 then 2^{a-1}\phi(m) = 2^{a-2}m\implies 2\phi(m) = m. This is impossible because LHS is even while RHS is odd.
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