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Math Help - Euler's totient function

  1. #1
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    Euler's totient function

    Find all positive integers n, such that \phi(n) is a prime.
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  2. #2
    Super Member PaulRS's Avatar
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    Quote Originally Posted by o_O View Post
    We must have that n is prime, since otherwise \phi (n) would have more than one prime factor.
    Careful, remember that \phi(2)=1. For example \phi(6)=2.

    To streethot: You should not have problems if you use the formula for the phi function (the one that uses the prime descomposition ), along with the observation o_O made that \phi(n) is either an even number, or the number 1.
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  3. #3
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    If \phi(n) is prime, then \varphi(n)=2, because if we have \varphi(n)= n\prod_{p|n} \left( 1-\frac{1}{p} \right), its easy to see that \prod_{p|n} \left( 1-\frac{1}{p} \right)\neq 1 for n\ge 2

    if \varphi(n)=(p_1 - 1) p_1^{k_1-1}...(p_n -1)p_n^{k_n-1} we have this inequality:

    (p_1 - 1) p_1^{k_1-1}...(p_n -1)p_n^{k_n-1}\leq(p_1 - 1) p_1^{k_1}...(p_n -1)p_n^{k_n} for \varphi(n)

    now, its possible to say that the n is in the form 2^k.3^m

    n=\{3,4,6\}

    Proceed?
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