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Thread: Proving integers?

  1. #1
    Aug 2012
    here and there

    Proving integers?

    Let n>1 be an integer. Prove that φ (n) | n if and only if n is of the form 2^a 3^b where a≥1 and b≥0 are integers.
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  2. #2
    Dec 2009

    Re: Proving integers?

    \varphi(2^a3^b)=2^a3^{b-1} | 2^a3^b

    \frac{n}{\varphi(n)} = \prod_{i=1}^{i=r}\frac{p_i}{p_i-1}
    the rhs denominator is dividable by 2^r if n is not dividable by 2 or 2^{r-1} if it is dividable by 2
    the rhs numerator is dividable only by 2 if n is dividable by a power of 2

    hence for the lhs to become an integer and thus \varphi(n)|n the number of primes in the factorization of n can be at most 2
    \frac{pq}{(p-1)(q-1)} must be integer

    therefore \frac{pq}{(p-1)(q-1)} \ge 2 equivalent to

    (*) 2 \ge (p-2)(q-2)

    wlog q<p

    examine the two cases:

    1) q=2: \frac{pq}{(p-1)(q-1)} = \frac{2p}{p-1} is integer only for p=3

    2) q \ge 3: then (*) becomes 2 \ge (p-2)(q-2) > p-2 thus p \le 4 since p prime p = 2 or 3 wich is a contradiction to q<p
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