[math-fun] Primes of the form 1+2*p^k
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?
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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Look at the paper of Erdos and Odlyzko: http://ac.els-cdn.com/0022314X7990043X/1-s2.0-0022314X7990043X-main.pdf?_tid... Sent from my iPhone
On Oct 24, 2015, at 14:09, James Buddenhagen <jbuddenh@gmail.com> 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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On Oct 24, 2015, at 12:55 PM, Henry Baker <hbaker1@pipeline.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) ?
"Primorial" primes. Hmmmm.. OK, here's a new(?) sequence for OEIS, perhaps called a "Goedel sequence" G(n): n in N maps to 2^d0*3^d1*5^d2*7^d3*11^d4*... d0 is 0'th bit of n; d1 is the 1st bit of n; d2 is the 2nd bit of n; d3 is the 3rd bit of n; etc. G(0) = 1 G(1) = 2 G(2) = 3 G(3) = 2*3 = 6 G(4) = 5 G(5) = 2*5 = 10 G(6) = 3*5 = 15 G(7) = 2*3*5 = 30 G(8) = 7 G(9) = 2*7 = 14 G(11) = 2*3*7 = 42 G(13) = 2*5*7 = 70 G(15) = 2*3*5*7 = 210 G(17) = 2*11 = 22 G(19) = 2*3*11 = 66 G(21) = 2*5*11 = 110 etc. A Goedelian prime would be an odd number n such that G(n)+1 is prime. G(1)+1 = 2+1 = 3 is prime G(3)+1 = 6+1 = 7 is prime G(5)+1 = 10+1 = 11 is prime G(7)+1 = 2*3*5+1 = 31 is prime G(9)+1 = 2*7+1 = 15 is NOT prime G(11)+1 = 2*3*7+1 = 43 is prime G(13)+1 = 2*5*7+1 = 71 is prime G(15)+1 = 2*3*5*7+1 = 211 is prime G(17)+1 = 2*11+1 = 23 is prime G(19)+1 = 2*3*11+1 = 67 is prime G(21)+1 = 2*5*11+1 = 111 is NOT prime etc. At 11:20 AM 10/24/2015, Hans Havermann wrote:
On Oct 24, 2015, at 12:55 PM, Henry Baker <hbaker1@pipeline.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) ?
I believe that http://oeis.org/A019565 corresponds to your "Goedel sequence". I assume you meant that a Goedelian prime is the prime G(n)+1 (not the odd number n). Including 2, aren't these just primes p such that p-1 is squarefree?
On Oct 24, 2015, at 5:18 PM, Henry Baker <hbaker1@pipeline.com> wrote:
OK, here's a new(?) sequence for OEIS, perhaps called a "Goedel sequence" G(n):
<deletia>
A Goedelian prime would be an odd number n such that G(n)+1 is prime.
Yes, A019565 does appear to be the same definition. I called it "Goedelian" ("Goedelial" ???) because Goedel used mappings to primes for his encodings, and it is sort of like Primorial... At 05:01 PM 10/24/2015, Hans Havermann wrote:
I believe that http://oeis.org/A019565 corresponds to your "Goedel sequence".
I assume you meant that a Goedelian prime is the prime G(n)+1 (not the odd number n).
Including 2, aren't these just primes p such that p-1 is squarefree?
On Oct 24, 2015, at 5:18 PM, Henry Baker <hbaker1@pipeline.com> wrote:
OK, here's a new(?) sequence for OEIS, perhaps called a "Goedel sequence" G(n):
<deletia>
The following is wrong; should be G(n)+1 itself, of course.
A Goedelian prime would be an odd number n such that G(n)+1 is prime.
participants (4)
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Hans Havermann -
Henry Baker -
James Buddenhagen -
Victor S. Miller