Let N have primitive roots .Which the way does find all primitive roots generally ? I think if testing all i ,(i,N)=1, is so long

THanks

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- May 13th 2008, 07:19 PMfanfan1609How to find a primitive root of N
Let N have primitive roots .Which the way does find all primitive roots generally ? I think if testing all i ,(i,N)=1, is so long

THanks - May 13th 2008, 08:14 PMThePerfectHacker
There is no known way to find primitive roots. There are methods which make it faster. Of course, a computer program is the best way to go. See this.

- May 13th 2008, 09:49 PMfanfan1609
I also know that way .YOu can help me my problem .

find the primitive root of 11^2 if we know 2,3 is primitive root of 11.

My teacher said that "if p is primitive root of N ,or p or p+N is primitive root of N^2 " but he didn't said the general primitive root of 11^2.

You can help me ?

Thanks - May 13th 2008, 10:08 PMThePerfectHacker
For 11 you can use a simple fact that if is a odd prime and is also a prime then is a primitive root for . I leave that for you to prove.

- May 13th 2008, 11:38 PMfanfan1609
- May 14th 2008, 07:59 AMThePerfectHacker
The direct way with to deal with is to list all with so . And now eliminate each of these numbers. For example, take . Check whether the order of mod is . Remember order means the

**smallest**exponent so that . But since (Fermat's theorem) it follows by properties of orders that so . Since it cannot be it means you just need to check . A simple calculation will show that is the order, so is not a primitive root. So that eliminates one number on the list . Now pick another number, say , using the approach as above you will find that . So that means is a primitive root mod . Now you can save yourself a lot of time, since you found one primitive root all other primitive roots are where and . In this case . Thus, are the primitive roots. - May 14th 2008, 04:17 PMfanfan1609
- May 14th 2008, 04:19 PMThePerfectHacker
- May 14th 2008, 04:27 PMfanfan1609
- May 14th 2008, 05:08 PMThePerfectHacker
- Jun 14th 2008, 08:42 AMduggaboy
Can you make a general statment or "rule" to determining if g^k is a primitive root? Is there a generalization?

- Jun 14th 2008, 09:01 AMfanfan1609
Sorry,I don't know your question.

If you want to determine a is a primitive root modulo n ,i think you should do something :

1/you find phi(n)

2/determine d with d|n

3/you test a^d mod n if the result isn't 1 with every d ,a is a primitive root modulo n

if you want find other primitive root ,you find e with e|phi(phi(n)) ,

with each e ,a^e mod n is a primitive root modulo n - Jun 14th 2008, 09:27 AMduggaboy
but how would you find d? just choose vales and try?

- Jun 14th 2008, 04:17 PMfanfan1609
- Jun 14th 2008, 06:46 PMThePerfectHacker