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Math Help - Euler Function phi(n)

  1. #16
    Newbie BBAmp's Avatar
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    Okay I was thinking it over and you are right I don't think I made any errors. I don't know why I used n/2. I think I was just in a hurry to write it out lol
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  2. #17
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    Quote Originally Posted by BBAmp View Post
    Okay I was thinking it over and you are right I don't think I made any errors. I don't know why I used n/2. I think I was just in a hurry to write it out lol

    Can anyone else confirm his results?
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  3. #18
    MHF Contributor chiph588@'s Avatar
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    Suppose  n is even and  n=2^am where  m is odd.

     \phi(2n)=\phi(2^{a+1}m)=\phi(2^{a+1})\phi(m) = 2^a\phi(m)

     2\phi(n)=2\phi(2^am)=2\phi(2^a)\phi(m)=2\cdot2^{a-1}\phi(m)=2^a\phi(m)

    Thus  \phi(2n)=2^a\phi(m) and  2\phi(n)=2^a\phi(m)\implies \phi(2n)=2\phi(n)
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  4. #19
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    Quote Originally Posted by chiph588@ View Post
    Suppose  n is even and  n=2^am where  m is odd.

     \phi(2n)=\phi(2^{a+1}m)=\phi(2^{a+1})\phi(m) = 2^a\phi(m)

     2\phi(n)=2\phi(2^am)=2\phi(2^a)\phi(m)=2\cdot2^{a-1}\phi(m)=2^a\phi(m)

    Thus  \phi(2n)=2^a\phi(m) and  2\phi(n)=2^a\phi(m)\implies \phi(2n)=2\phi(n)

    Okay so is his case for when n is odd valid?
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  5. #20
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by 1337h4x View Post
    Okay so is his case for when n is odd valid?
    This is for when n is even
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  6. #21
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    Quote Originally Posted by chiph588@ View Post
    This is for when n is even
    I know, but earlier a case was explained for when n was odd. Was the explanation that ensued sufficient and correct?
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  7. #22
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by 1337h4x View Post
    I know, but earlier a case was explained for when n was odd. Was the explanation that ensued sufficient and correct?
    The explanation I gave for when n is odd is a valid proof.
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  8. #23
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    Thank you!
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