# Thread: how to calculate EulerPhi( a+bI)

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

EulerPhi(3+6I)=20 why ?can some one give me detailed explanation , I would be greatful

2. ## Re: how to calculate EulerPhi( a+bI)

Originally Posted by wsc810
EulerPhi(3+6I)=20 why ?can some one give me detailed explanation , I would be greatful
What does this mean? What is I?

3. ## Re: how to calculate EulerPhi( a+bI)

Originally Posted by wsc810
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

$\displaystyle \chi$ $\displaystyle \sigma$

4. ## Re: how to calculate EulerPhi( a+bI)

I mean (a+bi) is Gaussian integer ，i not x

5. ## Re: how to calculate EulerPhi( a+bI)

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

6. ## Re: how to calculate EulerPhi( a+bI)

Originally Posted by wsc810
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 $\displaystyle |a+bi|=a^2+b^2$?

7. ## 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