Root 2 primes make up 38% of all prime numbers.

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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?




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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 at 17:59






  • 1




    "Root 2 primes": Do you mean primes for which 2 is a primitive root?
    – amWhy
    Jul 27 at 18:09






  • 3




    @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 at 18:33






  • 1




    Rosen refers to M.R. Murty's survey in Intelligencer from 1988. Check that out.
    – Jyrki Lahtonen
    Jul 27 at 18:35






  • 2




    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 at 19:04














up vote
4
down vote

favorite












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?




prime-numbers




share|cite|improve this question





















  • 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 at 17:59






  • 1




    "Root 2 primes": Do you mean primes for which 2 is a primitive root?
    – amWhy
    Jul 27 at 18:09






  • 3




    @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 at 18:33






  • 1




    Rosen refers to M.R. Murty's survey in Intelligencer from 1988. Check that out.
    – Jyrki Lahtonen
    Jul 27 at 18:35






  • 2




    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 at 19:04












up vote
4
down vote

favorite









up vote
4
down vote

favorite











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?




prime-numbers




share|cite|improve this question













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?





prime-numbers





share|cite|improve this question












share|cite|improve this question




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edited Jul 27 at 18:31
























asked Jul 27 at 17:56









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 at 17:59






  • 1




    "Root 2 primes": Do you mean primes for which 2 is a primitive root?
    – amWhy
    Jul 27 at 18:09






  • 3




    @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 at 18:33






  • 1




    Rosen refers to M.R. Murty's survey in Intelligencer from 1988. Check that out.
    – Jyrki Lahtonen
    Jul 27 at 18:35






  • 2




    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 at 19:04
















  • 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 at 17:59






  • 1




    "Root 2 primes": Do you mean primes for which 2 is a primitive root?
    – amWhy
    Jul 27 at 18:09






  • 3




    @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 at 18:33






  • 1




    Rosen refers to M.R. Murty's survey in Intelligencer from 1988. Check that out.
    – Jyrki Lahtonen
    Jul 27 at 18:35






  • 2




    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 at 19:04















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 at 17:59




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 at 17:59




1




1




"Root 2 primes": Do you mean primes for which 2 is a primitive root?
– amWhy
Jul 27 at 18:09




"Root 2 primes": Do you mean primes for which 2 is a primitive root?
– amWhy
Jul 27 at 18:09




3




3




@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 at 18:33




@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 at 18:33




1




1




Rosen refers to M.R. Murty's survey in Intelligencer from 1988. Check that out.
– Jyrki Lahtonen
Jul 27 at 18:35




Rosen refers to M.R. Murty's survey in Intelligencer from 1988. Check that out.
– Jyrki Lahtonen
Jul 27 at 18:35




2




2




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 at 19:04




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 at 19:04















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