Help with raising imaginary numbers to the power of a number

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JohnnyyB

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Verfasst am: 02.01.2016, 12:22 Titel: Help with raising imaginary numbers to the power of a number

f anyone could help, I would be so so so unbelievably thankful - so thank you for everyone reading it - you are great!

So here is the deal; I define a Gauß Sum with

$G(p,q)=\sum\limits_{m=1}^{q-1}\zeta_p^j*\zeta_q^m$

Here $\zeta_q=exp^{\frac{2 \pi i} {q} }$ and $\zeta_p=exp^{\frac{2 \pi i} {p} }$ and j that way, so that $g^j \equiv m \pmod{q}$ and $p \mid q-1$ (g is a already setted primitive root of q)

Now all Gauß Sums can be expressed uniquely as sums $\sum\limits_{j=0}^{p-2}\sum\limits_{k=0}^{q-2}a_{j,k}\zeta_p^j \zeta_q^k$

Now two elements $G(p,q)$ , $\overline{G}(p,q)$ of those ring are said to be congruent to each other modulo n if all $a_{j,k} \equiv \overline {a}_{j,k} \pmod{n}$ ...

Now my problem is to solve a problem like: Find the first positive integer u, so that:

$G(p,q)^u \equiv \zeta_p^j \pmod{n}$ for some integer j;

Also I try to implement this in Matlab;
So even if I manage to save the $a_{j,k}$ in the right way, when I raise it to the power of u and $u ] 1$ . Then I am truly unable how those $a_{j,k}$ develop like e.g. how the $a_{j,k}$ from $G(p,q)$ would develop if I calculate $G(p,q)^2$ . This is obviously still a part of the ring, so there must be integers $a_{j,k}$ also for this element but I have no idea how to get them when I really try to implement it
Anyone an idea for me? Smile

Jan S

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Verfasst am: 02.01.2016, 14:14 Titel: Re: Help with raising imaginary numbers to the power of a nu

Dear JohnnyyB,

I assume, that your question is far away from conering the Matlab implementation, but it sounds like the first steps are to obtain a mathematical method to find the solutions. Only if you have such a strategy, the implementation as code is possible.

Zitat:

Then I am truly unable how those $a_{j,k}$ develop like e.g. how the $a_{j,k}$ from $G(p,q)$ would develop if I calculate $G(p,q)^2$ .

I'm convinced that readers in this forum who are not working in exactly the same field of science as you, cannot reconsider, what you are talking of.

If you have tried to implement this partially in Matlab, and have a question concerning Matlab, please post the code.

Therefore I move this question to the Off-Topic category.

Kind regards, Jan

Verschoben: 02.01.2016, 14:14 Uhr von Jan S
Von Programmierung nach Off Topic


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