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Math Help - a few questions on (Z/nZ)* and Euler's Totient Function

  1. #1
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    a few questions on (Z/nZ)* and Euler's Totient Function

    phi(n) Euler's Totient Function:

    Given the group (Z/nZ)* would I be correct in saying that L = lcm(d1,...dk) where d1,...,dk are the distinct orders of the elements of (Z/nZ)* is the smallest integer for which a^L=1 for all elements of (Z/nZ)*?

    Secondly, since di | |(Z/nZ)*| = phi(n) for i=1,...,k am I correct in saying that L is less than or equal to phi(n)?

    Main question(s):

    When is phi(n)/L an integer (always?)? More generally, can we write phi(n)/L as f(phi(n)) some function of phi(n)? When is phi(n) an element of (Z/nZ)*?

    Cheers. Any help (on any of the questions) much appreciated.
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  2. #2
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    Re: a few questions on (Z/nZ)* and Euler's Totient Function

    Quote Originally Posted by b5345007 View Post
    phi(n) Euler's Totient Function:

    Given the group (Z/nZ)* would I be correct in saying that L = lcm(d1,...dk) where d1,...,dk are the distinct orders of the elements of (Z/nZ)* is the smallest integer for which a^L=1 for all elements of (Z/nZ)*?
    Yes.

    Secondly, since di | |(Z/nZ)*| = phi(n) for i=1,...,k am I correct in saying that L is less than or equal to phi(n)?
    Yes.

    Main question(s):

    When is phi(n)/L an integer (always?)?
    Yes, always.

    More generally, can we write phi(n)/L as f(phi(n)) some function of phi(n)?
    No. If we could then we would have 2=\varphi(8)/L_8=f(\varphi(8))=f(4)=f(\varphi(5))=\varphi(5)/L_5=1, a contradiction.

    When is phi(n) an element of (Z/nZ)*?
    \varphi(n) is an element of (\mathbb{Z}/n\mathbb{Z})^\times if and only if all of the following are false:

    (1) n=2r for r>2;

    (2) p^2\big|n for some prime p;

    (3) p divides (q-1)/2 for some odd prime divisors p,q of n.

    Cheers. Any help (on any of the questions) much appreciated.
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  3. #3
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    Re: a few questions on (Z/nZ)* and Euler's Totient Function

    Cheers!
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