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It has been conjectured by Emile Artin that the primes of primitive root 2 are infinite. Is it known that their density among all primes is about 1/3 and that this density is constant up to 1 quadrillion? I have a quick test for determining if primes are of primitive root two. I noticed that these "Artin primes" made up 1/3 (really about 38%) of primes up to 1000, 10,000 etc. up to 1Qd. Using a similar test, I found another class of primes that makes up another 27% of the density and is very constant as well. Is this a known result?

Casey Stewart
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  • Could you explain the context from which you conjecture the density among primes is 1/3, and that this density is constant up to quadrillion. Just provide a little background and work behind your conjecture. – amWhy Jul 27 '18 at 17:59
  • Possible duplicate of Are there infinitely many pairs of primes, $p$ and $q$, such that $q = 4p + 1$? – amWhy Jul 27 '18 at 18:04
  • This apparently is an open problem. So you want us to tell you whether the claim you right is true? Note that the question in the duplicate includes a reference to [this post](https://math.stackexchange.com/questions/256853/quadratic-reciprocity-and-proving-a-number-is-a-primitive-root – amWhy Jul 27 '18 at 18:05
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    "Root 2 primes": Do you mean primes for which 2 is a primitive root? – amWhy Jul 27 '18 at 18:09
  • yes. that is exactly what i am referring to. – Casey Stewart Jul 27 '18 at 18:16
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    @amWhy This is known as Artin's conjecture on primitive roots. If you knew about that I apologize. Anyway, M. Ram Murty et al have made what Rosen describes as great progress. According to WP this is known to hold under GRH. I guess that means there is a lot of evidence, but that is above my paygrade. The analogous fact in polynomial rings over a finite field is a theorem of Bilharz. GRH is known for that variant. – Jyrki Lahtonen Jul 27 '18 at 18:33
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    Rosen refers to M.R. Murty's survey in Intelligencer from 1988. Check that out. – Jyrki Lahtonen Jul 27 '18 at 18:35
  • Does Artin's conjecture cover primitive roots other than two? – Casey Stewart Jul 27 '18 at 18:36
  • Is it also known that primes having primitive root two are always congruent to either 3 or 5 (mod 8)? – Casey Stewart Jul 27 '18 at 18:40
  • Because the analogous class that I found with 27% density is always congruent to either 1 or 7 (mod 8). – Casey Stewart Jul 27 '18 at 18:41
  • I have a gorgeous color-coded Excel spreadsheet with the primes to 250,000 that demonstrates this density beautifully. – Casey Stewart Jul 27 '18 at 18:43
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    It would be helpful to state what class of primes you conjecture has density 27%, or else no one can respond as to its originality. – hardmath Jul 27 '18 at 19:04
  • I'm not sure how to define it. When we raise 2 to a power mod a prime and the residues cover every number up to that prime, the number is said to have primitive root two. What do we call it when the residues only cover exactly half of the numbers up to the prime? – Casey Stewart Jul 27 '18 at 19:06
  • The lowest examples are 5 and 7. For 5 we have (2, 4, 3, 1); but for seven we have (2, 4, 1) and then (6, 5, 3). The algorithm is novel but the class for which 7 is a member has a density of 27% among the primes up to 1Qd. – Casey Stewart Jul 27 '18 at 19:11
  • And, as stated in the question, the class for which 5 is a member has density approx. equal to 38%. – Casey Stewart Jul 27 '18 at 19:12
  • If there is a forum where we could chat about this, please inform me as I am new to stack. – Casey Stewart Jul 27 '18 at 19:38

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