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Math Help - how to calculate EulerPhi( a+bI)

  1. #1
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    Smile how to calculate EulerPhi( a+bI)

    EulerPhi(3+6I)=20 why ?can some one give me detailed explanation , I would be greatful
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: how to calculate EulerPhi( a+bI)

    Quote Originally Posted by wsc810 View Post
    EulerPhi(3+6I)=20 why ?can some one give me detailed explanation , I would be greatful
    What does this mean? What is I?
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    MHF Contributor chisigma's Avatar
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    Re: how to calculate EulerPhi( a+bI)

    Quote Originally Posted by wsc810 View Post
    EulerPhi(3+6I)=20 why ?can some one give me detailed explanation , I would be greatful
    Remembering that is...

    \varphi(n) = n\ \prod_{p|n} (1-\frac{1}{p}) (1)

    ... first we search the n for which is \varphi(n)=20. Searching the primes p_{i} so that (p_{i}-1)|20 we find p_{1}=2, p_{2}=3, p_{3}=5 and p_{4}=11, so that n must contain one or more of these primes and necessarly n>20. Taking into account that we find that the possible values of n are n=25, n=33, n=50 and n=66. Among these n, the only for which is 3+6\ i = n is n=33 so that is i=5...

    Kind regards

    \chi \sigma
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    Re: how to calculate EulerPhi( a+bI)

    I mean (a+bi) is Gaussian integer ,i not x
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  5. #5
    MHF Contributor chisigma's Avatar
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    Re: how to calculate EulerPhi( a+bI)

    All right!... before trying to answer You only a little question: where did You read that \varphi(3+i\ 6)=20?...

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    \chi \sigma
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    MHF Contributor Drexel28's Avatar
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    Re: how to calculate EulerPhi( a+bI)

    Quote Originally Posted by wsc810 View Post
    I mean (a+bi) is Gaussian integer ,i not x
    That doesn't make sense. I have never heard of the phi function extended to \mathbb{Z}[i]. How do you define it, \varphi(a+bi) is the number of coprime elements of \mathbb{Z}[i] whose modulus is less than |a+bi|=a^2+b^2?
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  7. #7
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    Re: how to calculate EulerPhi( a+bI)

    Yes,it is .if you know chinese ,you could read the book 《algebraic number theory》which Pan chengdong write ,but I don't understand,it's deep for me
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