Re: primitive root error

*To*: mathgroup at smc.vnet.net*Subject*: [mg128094] Re: primitive root error*From*: Bob Hanlon <hanlonr357 at gmail.com>*Date*: Sat, 15 Sep 2012 03:37:59 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <20120914042435.8DEC1684C@smc.vnet.net>

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 > answer is 5: > > In[1]:= Table[Mod[5^n, 18], {n, 0, 6}] > > Out[1]= {1, 5, 7, 17, 13, 11, 1} >

**References**:**primitive root error***From:*Dan Dubin <ddubin@ucsd.edu>