Re: [math-fun] Primes of the form 1+2*p^k
Thanks! p=1 (mod 3) also kills off 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, etc. p^2=1 (mod 3) [p=2=-1 (mod 3)] kills off even powers of the other primes /= 3. Hans Havermann suggests: http://oeis.org/A005234 'Primorial primes' (I'd never heard of this term before) are interesting, but I'm also allowing any subsequence of the primes, so long as they contain 2. At 11:09 AM 10/24/2015, James Buddenhagen wrote:
1+2*7^k is always divisible by 3
On Sat, Oct 24, 2015 at 11:55 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Dumb questions re primes.
Primes of the form 1+2^k are quite rare. ;-)
Primes of the form 1+2*3^k seem to be less rare.
Primes of the form 1+2*5^k seem to get rarer.
Primes of the form 1+2*7^k seem to be quite rare. (I don't have a fast machine, but I'm having trouble finding even one.)
Primes of the form 1+2*11^k seem to be less rare.
Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are odd primes (i.e., primes to the 1st power only) ?
(Perhaps these primes should be called "Euclid primes" after Euclid's proof of the infinite # of primes -- if they have no other name?)
Anything known about these distributions?
Also, is the discrete log particularly cheap to compute for any of these prime forms?
Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are odd primes (i.e., primes to the 1st power only) ?
This would be (essentially) https://oeis.org/A039787 Click "graph" to get an idea of how fast they grow Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, Oct 24, 2015 at 2:37 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Thanks!
p=1 (mod 3) also kills off 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, etc.
p^2=1 (mod 3) [p=2=-1 (mod 3)] kills off even powers of the other primes /= 3.
Hans Havermann suggests:
'Primorial primes' (I'd never heard of this term before) are interesting, but I'm also allowing any subsequence of the primes, so long as they contain 2.
At 11:09 AM 10/24/2015, James Buddenhagen wrote:
1+2*7^k is always divisible by 3
On Sat, Oct 24, 2015 at 11:55 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Dumb questions re primes.
Primes of the form 1+2^k are quite rare. ;-)
Primes of the form 1+2*3^k seem to be less rare.
Primes of the form 1+2*5^k seem to get rarer.
Primes of the form 1+2*7^k seem to be quite rare. (I don't have a fast machine, but I'm having trouble finding even one.)
Primes of the form 1+2*11^k seem to be less rare.
Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are odd primes (i.e., primes to the 1st power only) ?
(Perhaps these primes should be called "Euclid primes" after Euclid's proof of the infinite # of primes -- if they have no other name?)
Anything known about these distributions?
Also, is the discrete log particularly cheap to compute for any of these prime forms?
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If I understand correctly these primes 'should' have density A = 37.39...%, where A is Artin's constant. Ah! I found a proof by Mirsky, who cites Estermann: Theodor Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105:1 (1931), pp. 653-662. I added both to https://oeis.org/A039787 Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Oct 25, 2015 at 3:40 AM, Neil Sloane <njasloane@gmail.com> wrote:
Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are odd primes (i.e., primes to the 1st power only) ?
This would be (essentially) https://oeis.org/A039787
Click "graph" to get an idea of how fast they grow
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sat, Oct 24, 2015 at 2:37 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Thanks!
p=1 (mod 3) also kills off 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, etc.
p^2=1 (mod 3) [p=2=-1 (mod 3)] kills off even powers of the other primes /= 3.
Hans Havermann suggests:
'Primorial primes' (I'd never heard of this term before) are interesting, but I'm also allowing any subsequence of the primes, so long as they contain
At 11:09 AM 10/24/2015, James Buddenhagen wrote:
1+2*7^k is always divisible by 3
On Sat, Oct 24, 2015 at 11:55 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Dumb questions re primes.
Primes of the form 1+2^k are quite rare. ;-)
Primes of the form 1+2*3^k seem to be less rare.
Primes of the form 1+2*5^k seem to get rarer.
Primes of the form 1+2*7^k seem to be quite rare. (I don't have a
fast
machine, but I'm having trouble finding even one.)
Primes of the form 1+2*11^k seem to be less rare.
Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are odd primes (i.e., primes to the 1st power only) ?
(Perhaps these primes should be called "Euclid primes" after Euclid's proof of the infinite # of primes -- if they have no other name?)
Anything known about these distributions?
Also, is the discrete log particularly cheap to compute for any of these prime forms?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Charles Greathouse -
Henry Baker -
Neil Sloane