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Prime Constellations Research Project
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Thomas R. Nicely

Freeware copyright © 2010 Thomas R. Nicely. Released into the public
domain by the author, who disclaims any legal liability arising from
its use.
Last updated 1930 GMT 16 March 2010.

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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 ± ??

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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