Re: primitive root error

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• Subject: [mg128094] Re: primitive root error
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Sat, 15 Sep 2012 03:37:59 -0400 (EDT)
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Your example does not appear to be an isolated case

PrimitiveRoot[26]

15

Table[Mod[15^n, 26], {n, 0, 12}]

{1, 15, 17, 21, 3, 19, 25, 11, 9, 5, 23, 7, 1}

Union[%] == Union[Table[Mod[7^n, 26], {n, 0, 12}]]

True

PrimitiveRoot[58]

31

Table[Mod[31^n, 58], {n, 0, 28}]

{1, 31, 33, 37, 45, 3, 35, 41, 53, 19, 9, 47, 7, 43, 57, 27, 25, 21, 13, 55, \
23, 17, 5, 39, 49, 11, 51, 15, 1}

Union[%] == Union[Table[Mod[15^n, 58], {n, 0, 28}]]

True

Table[Prime[n], {n, 15}]

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}

PrimitiveRoot[74]

39

Table[Mod[39^n, 74], {n, 0, 36}]

{1, 39, 41, 45, 53, 69, 27, 17, 71, 31, 25, 13, 63, 15, 67, 23, 9, 55, 73, \
35, 33, 29, 21, 5, 47, 57, 3, 43, 49, 61, 11, 59, 7, 51, 65, 19, 1}

Union[%] == Union[Table[Mod[19^n, 74], {n, 0, 36}]]

True

PrimitiveRoot[82]

47

Table[Mod[47^n, 82], {n, 0, 40}]

{1, 47, 77, 11, 25, 27, 39, 29, 51, 19, 73, 69, 45, 65, 21, 3, 59, 67, 33, \
75, 81, 35, 5, 71, 57, 55, 43, 53, 31, 63, 9, 13, 37, 17, 61, 79, 23, 15, 49, \
7, 1}

Union[%] == Union[Table[Mod[19^n, 82], {n, 0, 40}]]

True

Bob Hanlon

On Fri, Sep 14, 2012 at 12:24 AM, Dan Dubin <ddubin at ucsd.edu> wrote:
> The number theoretic function PrimitiveRoot[n]  is supposed to give the
> smallest generator for the multiplicative group of integers module n
> relatively prime to n.  However, Mathematica 8  says that
> PrimitiveRoot[18] equals 11. This is incorrect. While this is a
> generator, it is not the smallest generator of the group. The correct
>
> In[1]:= Table[Mod[5^n, 18], {n, 0, 6}]
>
> Out[1]= {1, 5, 7, 17, 13, 11, 1}
>

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