Prime Constellations Research Project
Thomas R. Nicely
Freeware copyright (c) 2009 Thomas R. Nicely. Released into the public
domain by the author, who disclaims any legal liability arising from
its use.
Last updated 0642 GMT 5 October 2009.
TWIN PRIMES
Stated below, for the present upper bound of the author's
computations, are the count pi_2 of twin-prime pairs (q, q+2);
the partial sum S_2 of the reciprocals of the
twins; the resulting extrapolated estimate for Brun's constant
B_2; and an error estimate which the author conjectures
to represent at least one standard deviation (a 68.27 % confidence
level). For details of the error analysis, see the papers listed
below.
pi_2(2e16) = 19,831,847,025,792
S_2(2e16) = 1.83180 80634 32379 01198 41239 12086 74712 537...
B_2 = 1.90216 05832 09 ± 0.00000 00007 81
PRIME TRIPLETS (q, q+2, q+6)
Stated below, for the present upper bound of the author's
computations, are the count pi_3a of the prime triplets (q, q+2, q+6);
the partial sum S_3a of the reciprocals of these triplets;
and the resulting extrapolated estimate for the corresponding Brun's
constant B_3a. The analysis for the error estimate is in progress,
and will be posted when available.
pi_3a(2e16) = 1,178,112,426,442
S_3a(2e16) = 1.09480 78446 39407 97537 62763 85969 0454...
B_3a = 1.09785 10396 79 ± ??
PRIME TRIPLETS (q, q+4, q+6)
Stated below, for the present upper bound of the author's
computations, are the count pi_3b of the prime triplets (q, q+4, q+6);
the partial sum S_3b of the reciprocals of these triplets;
the resulting extrapolated estimate for the corresponding Brun's
constant B_3b. The analysis for the error estimate is in progress,
and will be posted as available.
pi_3b(2e16) = 1,178,110,447,049
S_3b(2e16) = 0.83407 00173 71509 54996 95378 70284 26700 4436...
B_3b = 0.83711 32124 11 ± ??
PRIME QUADRUPLETS
Stated below, for the present upper bound of the author's
computations, are the count pi_4 of prime quadruplets (q, q+2, q+6, q+8);
the partial sum S_4 of the reciprocals of the quadruplets;
the resulting extrapolated estimate for the corresponding Brun's constant
B_4; and an error estimate which the author conjectures
to represent at least one standard deviation (a 68.27 % confidence
level). For details of the error analysis, see the paper
"Enumeration to 1.6e15 of the
prime quadruplets".
pi_4(2e16) = 46,998,268,431
S_4(2e16) = 0.87048 37109 48052 51495 90538 79648 11225 58492 6...
B_4 = 0.87058 83799 75 ± 0.00000 00001 14
- Top of page
- A new error analysis for Brun's
constant (paper, 2001)
- Enumeration to 1.6e15 of the twin
primes and Brun's constant (paper, 1999)
- Enumeration to 1e14 of the twin primes
and Brun's constant (paper, 1995)
- Enumeration to 1.6e15 of the
prime quadruplets (paper, 1999)
- Enumeration of the twin-prime pairs
to 1e16 (table)
- Enumeration of the twin-prime pairs
from 1e16 to 2e16 (table)
- Enumeration of the prime quadruplets
to 1e16 (table)
- Enumeration of the prime quadruplets
from 1e16 to 2e16 (table)
- Enumeration of the prime triplets
(q, q+2, q+6) to 1e16 (table)
- Enumeration of the prime triplets
(q, q+2, q+6) from 1e16 to 2e16 (table)
- Enumeration of the prime triplets
(q, q+4, q+6) to 1e16 (table)
- Enumeration of the prime triplets
(q, q+4, q+6) from 1e16 to 2e16 (table)
- Home page