Artin's ConjectureDate: 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/ |
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