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Re: primitive root error

• To: mathgroup at smc.vnet.net
• Subject: [mg128096] Re: primitive root error
• From: daniel.lichtblau0 at gmail.com
• Date: Sat, 15 Sep 2012 03:38:39 -0400 (EDT)
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On Thursday, September 13, 2012 11:16:47 PM UTC-5, Dan Dubin 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}

Me, I'd call that a mistake in the documentation. Anyway, you can get the smallest by resorting to a crude, nay, barbaric tactic: invoke Wolfram|Alpha to find all the primitive roots.

In[71]:= WolframAlpha["primitive roots of 18", "MathematicaResult"]

Out[71]= {5, 11}

Daniel Lichtblau
Wolfram Research

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