Artin's Conjecture

Date: 07/11/2002 at 11:34:36
From: Lekraj Beedassy
Subject: Number Theory

Hi Dr. Math,

Examining tables of primitive roots of primes, and of numbers in
general, it soon becomes clear that some numbers never show up as
primitive roots. An empirical deduction of mine is that prime powers
can never be primitive roots; a bolder assertion still would be that
perfect powers can never be primitive roots.

How far am I right in my conjectures? Which numbers really can never
be primitive roots and how do we prove that?

Thank you in advance for at least providing a clue.

L.B.

Date: 07/12/2002 at 12:03:13
From: Doctor Nitrogen
Subject: Re: Number Theory

Hello, L.B.,

The reason some numbers do not show up in your table for primitive 
roots modulo p (p a prime) is that for any prime p, there are only  
phi(p) = p - 1 primitive roots modulo p. Here "phi" denotes Euler's 
Totient Function.

For which primes p is a given number a primitive root?  Even the 
Primitive Root Theorem does not indicate whether a given number, a, is 
a primitive root. So this remains right now an open question.

There is, however, a Conjecture by Artin, which states the following:

  Artin's Conjecture: Let a be any integer not a perfect 
  square and not equal to -1. Then there are infinitely 
  many primes p such that a is a primitive root modulo p. 

This conjecture has not been proved yet. 

I hope this was helpful. Return to Dr. Math with more questions if 
you have any.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/