Thomas R. Nicely
Last updated 2300 EDT 19 March 2008.
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
"Enumeration to $1.6 \times 10^{15}$
of the twin primes and 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
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