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Math Help - Nontotient Number

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

    A nontotient number  n \in \mathbb{N} is a number such that there is no  x \in \mathbb{N} where  \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  n \in \mathbb{N} is a number such that there is no  x \in \mathbb{N} where  \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 3. Because \phi (x) is always even for x>2.
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  3. #3
    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  n \in \mathbb{N} is a number such that there is no  x \in \mathbb{N} where  \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 2,4,6,8,10,12 are all non-nontotient.
    After that you need to show that 14 is a non-totient number.

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