Prime Constellations Research Project
Thomas R. Nicely
Copyright © 2008 Thomas R. Nicely. All rights reserved.
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Last updated 1001 GMT 22 September 2008.
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 be definitive. For details of the error analysis, see the paper
"A new error analysis for Brun's
constant."
pi_2(1e16) = 10,304,195,697,298
S_2(1e16) = 1.83048 44246 58338 48374 01122 82692 11302...
B_2 = 1.90216 05831 05 ± 0.00000 00011 25
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 be definitive. For details of the error analysis, see the paper
"Enumeration to $1.6 \times 10^{15}$
of the prime quadruplets"; further details will appear in the
paper "Enumeration of the prime quadruplets (q, q+2, q+6, q+8)"
(in preparation).
pi_4(1e16) = 25,379,433,651
S_4(1e16) = 0.87047 76912 34045 95410 35953 13883 51533 372...
B_4 = 0.87058 83799 57 ± 0.00000 00001 68
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 as available.
pi_3a(1e16) = 624,026,299,748
S_3a(1e16) = 1.09469 22555 99340 38266 03614 98767 18197 1754...
B_3a = 1.09785 10394 95 ± ??
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(1e16) = 624,025,508,307
S_3b(1e16) = 0.83395 44285 65136 97017 17810 08079 35124 9546...
B_3b = 0.83711 32124 60 ± ??