Re: primitive root error

*To*: mathgroup at smc.vnet.net*Subject*: [mg128099] Re: primitive root error*From*: Andrzej Kozlowski <akozlowski at gmail.com>*Date*: Sat, 15 Sep 2012 03:39:39 -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>

I think it is just bad documentation. It seems that PrimitiveRoot[n] gives the smallest primitive root when n is a power of an odd prime. However, when g=PrimitiveRoot[p^k] is odd (where p is an odd prime) PrimitiveRoot[2 p^k] returns g+2^k, which is usually not the smallest primitive root. For example: PrimitiveRoot[41^2] 6 which is even. So now: PrimitiveRoot[2*41^2] 1687 which is 6+41^2. But the smallest PrimitiveRoot is: First[ Select[Range[2*41^2], MultiplicativeOrder[#, 2*41^2] == EulerPhi[2*41^2] &]] 7 which is a lot smaller. Andrzej Kozlowski On 14 Sep 2012, at 06:24, 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>