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Thread: 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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    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...

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

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

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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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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    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 $\displaystyle \varphi(3+i\ 6)=20$?...

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    $\displaystyle \chi$ $\displaystyle \sigma$
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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 $\displaystyle \mathbb{Z}[i]$. How do you define it, $\displaystyle \varphi(a+bi)$ is the number of coprime elements of $\displaystyle \mathbb{Z}[i]$ whose modulus is less than $\displaystyle |a+bi|=a^2+b^2$?
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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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