Results 1 to 3 of 3

Math Help - Euler's phi function: multiplicity

  1. #1
    Newbie
    Joined
    May 2009
    Posts
    8

    Euler's phi function: multiplicity

    I am supposed to show that, for Euler's totient function:

    a.) If n is an odd integer, then \phi(2n)=\phi(n).
    and
    b.) If n is an even integer, then \phi(2n)=2\phi(n).

    I thought I would use the multiplicity of phi to establish this, i.e.
    \phi(2n)=\phi(2)\cdot\phi(n)=1\cdot\phi(n)=\phi(n).

    Obviously this doesn't work, because while it works for (a), it contradicts (b). Not sure why.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    May 2009
    Posts
    8
    Sorry, title should be "multiplicativity", not "multiplicity".
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member TheAbstractionist's Avatar
    Joined
    Apr 2009
    Posts
    328
    Thanks
    1
    Quote Originally Posted by glowplug19 View Post
    I am supposed to show that, for Euler's totient function:

    a.) If n is an odd integer, then \phi(2n)=\phi(n).
    and
    b.) If n is an even integer, then \phi(2n)=2\phi(n).

    I thought I would use the multiplicity of phi to establish this, i.e.
    \phi(2n)=\phi(2)\cdot\phi(n)=1\cdot\phi(n)=\phi(n).

    Obviously this doesn't work, because while it works for (a), it contradicts (b). Not sure why.
    Hi glowplug19.

    Be careful here. The formula \phi(mn)=\phi(m)\phi(n) only works when \gcd(m,n)=1. This is why it works for (a) but not for (b), as \gcd(2,n)=1 if and only if n is odd.

    For (b) consider S_1=\{1,2\ldots,n\} and S_2=\{n+1,n+2\ldots,n+n=2n\}. If n is even, the integers in S_1 which are coprime with n excludes the even integers; hence an integer in S_1 is coprime with 2n if and only if it is coprime with n. For S_2 note that n+r is coprime with n if and only if r is coprime with n; hence the same argument can be applied to S_2 to show that there are \phi(n) integers in S_2 coprime with 2n. Thus \phi(2n)=\phi(n)+\phi(n)=2\phi(n).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Euler phi-function
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 15th 2010, 03:38 PM
  2. Euler Phi function II
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: December 27th 2009, 05:33 AM
  3. Again, phi euler function
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 22nd 2009, 12:35 PM
  4. Euler's phi function
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: November 16th 2008, 06:54 PM
  5. Euler's phi function
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: July 29th 2008, 05:07 AM

Search Tags


/mathhelpforum @mathhelpforum