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Math Help - euler fhi function and power of a prime

  1. #1
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    euler fhi function and power of a prime

    if (n-1) is divisible by φ(n) then there is no prime p such that p^2|m.

    Thanks...
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  2. #2
    Super Member PaulRS's Avatar
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    Re: euler fhi function and power of a prime

    I suppose you meant to say p^2 | n .

    Let's proceed by contradiction, suppose on the contrary that there were a prime p such that p^2 | n. Clearly p\not | (n-1) since n and (n-1) are coprime, however p^2 | n implies p | \varphi ( n) (*), and since \varphi(n) | (n-1) we have that p | (n-1) which is a contradiction. \square

    (*) Check this by using the formula \phi(n) = n \cdot \prod_{p|n} \left( 1 - \frac{1}{p}  \right) , where the products runs over all primes dividing n.
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  3. #3
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    Re: euler fhi function and power of a prime

    Thank you Paul for your nice proof...
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