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Math Help - A proof using the phi-function

  1. #1
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    A proof using the phi-function

    Given a positive integer k, show there are at most a finite number of integers n for which Φ(n)=k
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  2. #2
    Senior Member
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    Bounding the function

    It is already known that the totient function is bounded as follows:

    \sqrt{n} < \phi(n) < n-\sqrt{n}

    So, given an arbitrary positive integer k, \phi(n)>k for all n>k^2. So the highest number n satisfying \phi(n)=k must still be less than k^2. Since there is such an upper bound, the total number of integers n satisfying this equality is finite.
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