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Thread: Nontotient Number

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Nontotient Number

    A nontotient number $\displaystyle n \in \mathbb{N} $ is a number such that there is no $\displaystyle x \in \mathbb{N} $ where $\displaystyle \phi(x) = n $.

    The smallest such number is 14.

    My question is how would one prove whether a number is nontotient or not?
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    Quote Originally Posted by chiph588@ View Post
    A nontotient number $\displaystyle n \in \mathbb{N} $ is a number such that there is no $\displaystyle x \in \mathbb{N} $ where $\displaystyle \phi(x) = n $.

    The smallest such number is 14.

    My question is how would one prove whether a number is nontotient or not?
    I disagree. The smallest number is $\displaystyle 3$. Because $\displaystyle \phi (x)$ is always even for $\displaystyle x>2$.
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    MHF Contributor chiph588@'s Avatar
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    whoops, i forgot to mention the trivial answer of odd numbers...
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    Quote Originally Posted by chiph588@ View Post
    A nontotient number $\displaystyle n \in \mathbb{N} $ is a number such that there is no $\displaystyle x \in \mathbb{N} $ where $\displaystyle \phi(x) = n $.

    The smallest such number is 14.

    My question is how would one prove whether a number is nontotient or not?
    You first need to show that $\displaystyle 2,4,6,8,10,12$ are all non-nontotient.
    After that you need to show that $\displaystyle 14$ is a non-totient number.

    We want $\displaystyle n$ so that $\displaystyle \phi (n) = 14$.
    Write $\displaystyle n = p_1^{a_1}\cdot ... \cdot p_m^{a_m}$.
    Then we have $\displaystyle \phi (n) = p_1^{a_1-1} \cdot ... \cdot p_m^{a_m - 1}(p_1-1)(p_2 - 1)...(p_m -1) = 2\cdot 7$.
    The RHS has only one factor of $\displaystyle 2$.
    Therefore we cannot have $\displaystyle m\geq 2$ where $\displaystyle p_i$ are odd primes.
    The RHS has also a factor of $\displaystyle 7$.
    This forces $\displaystyle n=2^a 7^b$ where $\displaystyle a=0,1$ and $\displaystyle b=0,1$.
    This never works to give $\displaystyle \phi(n) = 14$.
    Thus, $\displaystyle 14$ is nontotient.
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