Enumeration to 1.6*10^15
of the twin primes and Brun's constant
Thomas R. Nicely
Copyright © 2007 Thomas R. Nicely. All rights reserved.
This document may be freely reproduced and distributed for
educational and non-profit purposes.
Date of first release 16 July 1999.
Last revised 0430 EDT 28 March 2007.
The content of this document (other than the addendum, which was
not part of the submission for publication) is essentially that of
the original release, except that information rendered obsolete by
subsequent events has been removed or modified, in both the main
document and the addendum (this liberty is taken in view of the
fact that the paper was never accepted for publication). There may
also be differences in formatting, and in minor details and
corrections.
Abstract
Enumeration of the twin primes, and the sum of their reciprocals,
is extended to 1.6*10^15. An improved estimate is obtained for
Brun's constant, B_2 = 1.90216 05824 +/- 0.00000 00030. Error
analysis is presented to support the contention that the stated
error bound represents a 99 % confidence level.
Mathematics Subject Classification 2000 (MSC2000)
Primary: 11A41.
Secondary: 11-04, 11Y70, 11Y60.
Key Words and Phrases
Twin primes, Brun's constant, prime numbers.
1. Introduction
The set K_2 = {(3, 5), (5, 7), (11, 13), (17, 19), ...} of twin-prime
pairs (q, q+2) has been studied by many investigators, including
Glaisher [7], Brun [3], Hardy and Littlewood [9], Sutton [22],
Selmer [18], Sexton [19], Lehmer [11], Fröberg [4],
Gruenberger and Armerding [8], Weintraub [23], Bohman [1],
Shanks and Wrench [20], Brent [2], Nicely [13], and Kutrib and
Richstein [10, 25]. Computations similar to those described here are
currently being carried out by Sebah [17] and by Fry, Nesheiwat,
and Szymanski [6].
The present study results from the continuation of a project initiated
in 1993, with results to 10^14 described in the author's
1995 paper [13]. A detailed description of the general problem, the
computational methods employed, and the incidental discovery of the
Pentium FDIV flaw may be found there, with additional details
given in [14] and [15]; only a brief summary will
be included here.
The prime numbers themselves continue to retain most of their secrets,
but still less is known about the twin primes. A matter as fundamental
as the infinitude of K_2 remains undecided---the famous
"twin-primes conjecture," a topic of discussion even in a recent
popular motion picture [21]. Nonetheless, Brun [3] proved in 1919
that in any event the sum of the reciprocals,
(1.1)
B_2 = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + ... ,
is convergent, in contrast to the known divergence of the sum of the
reciprocals of all the primes. The limit of this sum, styled Brun's
sum or Brun's constant, is usually denoted by simply B (although
Fröberg [4] used S), but herein the author will use
B_2. In this instance, as in a number of others noted in this
paper, identifiers have been changed from those in [2]
and [13], in anticipation of the need for analogous symbols to
be used in the study of prime constellations other than the twins.
The twin-primes conjecture is a consequence of a much stronger
result proposed by Hardy and Littlewood [9] in 1923,
(1.2)
pi_2(x) ~ L_2(x) = 2c_2*int(1/(ln(t))^2, t, 2, x)
where pi_2(x) represents the count of twin-prime pairs
(q, q+2) such that q <= x, and c_2 denotes the
"twin-primes constant," computed to 42D by Wrench [24] in 1961,
(1.3)
c_2 = 0.66016 18158 46869 57392 78121 10014 55577 84326 23...
The validity of the conjecture (1.2), sometimes titled the
Hardy-Littlewood approximation, is central to the estimation
of Brun's constant and the error bounds in this paper. The
Hardy-Littlewood approximation is itself a special case of the
yet more general "prime k-tuples conjecture," also set forth
in [9]. See Riesel [16, pp. 60-83] for an illuminating
exposition of these concepts.
Although (1.1) is convergent, the monotonically increasing
partial sums approach the limit with agonizing slowness; summing the
first thousand million reciprocals is still insufficient to bring us
within five percent of the estimated value of the limit. However,
assuming the validity of the Hardy-Littlewood approximation (1.2),
a first order extrapolation was derived by Fröberg [4] and
further studied by Brent [2],
(1.4)
B_2 = S_2(x) + 4c_2/ln(x) + O(1/(sqrt(x)*ln(x))
with an accelerated rate of convergence O(sqrt(x)) faster than
(1.1). Here S_2(x) is the partial sum
(1.5)
S_2(x) = sum(1/q + 1/(q+2), q, q <= x) .
of the reciprocals of all the twin-prime pairs (q, q+2) for which
q <= x. Note that S_2(x) is written as B(x) in [2]
and [13]. The first order extrapolation of S_2(x) to
approximate B_2 consists of the first two terms of the
right hand side of (1.4); this was indicated as B*(x)
in [2] and [13], but we write it here as F_2(x):
(1.6) F_2(x) = S_2(x) + 4c_2/ln(x) .
The final term in (1.4) is the author's conjectured error
or remainder term, inspired by Brent's probabilistic analysis [2].
As discussed by Shanks and Wrench [20, p. 298], no effective
second-order extrapolation is known.
2. Computational Technique
These calculations were carried out as part of a more comprehensive
project, including in addition enumerations of other prime
constellations and prime gaps. Computations began in 1993 and have
since proceeded almost without interruption, although several months'
work was lost early on due to the Pentium FDIV flaw. The
calculations were distributed across a number (varying from a few to
more than two dozen) of personal computers, using Intel
processors (mostly classic Pentiums), extended DOS and
Windows 3.x operating systems, and code written in C. The
algorithm employed the classic sieve of Eratosthenes to carry out
an exhaustive generation and enumeration of the primes. To guard
against errors, all calculations were performed in duplicate on
separate systems; in addition, the count pi(x) of primes was
maintained and checked periodically against known values. The values
obtained for the count pi_2(x) of twin-prime pairs agreed to
10^11 with those of Brent [2], and to 10^14 with those
of Kutrib and Richstein [10, 25]. Additional values of pi_2(x)
to 8*10^14 were checked with preliminary values obtained
by Fry, Nesheiwat, and Szymanski [5]. In addition to the
errors generated by the Pentium FDIV flaw, more than fifty
other errors have been detected, most apparently the result
of faults---some transient, some permanent---in RAM chips.
The reciprocal sums were calculated in both long double precision (64
bit mantissa, 19S) and ultraprecision (53D), the former in hardware
using the Intel numeric coprocessors, the latter in software
using a modification of a package due largely to Arjen Lenstra.
Results were written to a binary file at intervals of 10^10.
As mentioned previously, additional details regarding the computational
technique, and the Pentium FDIV affair, are available in the cited
references [13, 14, 15], and also at the author's URL.
3. Computational Results
Table 1 contains a brief summary of the computational results,
including the counts pi_2(x) of twin-prime pairs; the values of
the discrepancy between pi_2(x) and the Hardy-Littlewood
approximation, indicated here by delta_2(x),
(3.1) delta_2(x) = L_2(x) - pi_2(x) ;
the partial sums S_2(x) of the reciprocals of the twins; and the
first order extrapolations F_2(x) of S_2(x)
to the limit, according to (1.6), members of a sequence believed
to be converging to Brun's constant B_2. Note that the
discrepancy delta_2(x) was written in [2] and [13] as
r_3(x); Brent also rounded this value to the nearest integer.
Table 1 includes results previously published for powers
of ten to 10^14, in addition to new results from the present
study for x = (1.5*10^14)(5*10^13)(1.6*10^15). Expanded
versions of Table 1 are available at the author's URL, as are
extensive tables of the values of pi(x) recorded in this project,
at much finer granularity than those commonly available.
Table 1. Counts of twin-prime pairs and estimates of Brun's constant.
=======================================================================
x pi_2(x) delta_2(x) S_2(x) F_2(x)
=======================================================================
10 2 2.84 0.8761904761905 2.0230090113326
100 8 5.54 1.3309903657191 1.9043996332901
1000 35 10.80 1.5180324635596 1.9003053086070
10000 205 9.21 1.6168935574322 1.9035981912177
10^05 1224 24.71 1.6727995848277 1.9021632918562
10^06 8169 79.03 1.7107769308042 1.9019133533279
10^07 58980 -226.18 1.7383570439173 1.9021882632233
10^08 440312 55.79 1.7588156210680 1.9021679379607
10^09 3424506 802.16 1.7747359576385 1.9021602393210
10^10 27412679 -1262.47 1.7874785027192 1.9021603562335
10^11 224376048 -7183.32 1.7979043109551 1.9021605414226
10^12 1870585220 -25353.18 1.8065924191759 1.9021606304377
10^13 15834664872 -66566.94 1.8139437606846 1.9021605710802
10^14 135780321665 -56770.51 1.8202449681303 1.9021605777833
1.50e14 198466021764 -41424.45 1.8212624999372 1.9021605778478
2.00e14 259858400254 -286596.19 1.8219692563019 1.9021605806674
2.50e14 320314774824 -251822.16 1.8225090097373 1.9021605803406
3.00e14 380041003032 -386165.49 1.8229446574499 1.9021605813179
3.50e14 439169847802 -377896.90 1.8233092906771 1.9021605812270
4.00e14 497794845572 -687458.42 1.8236224494488 1.9021605828234
4.50e14 555983876545 -538028.53 1.8238966154614 1.9021605820874
5.00e14 613790177314 -495402.94 1.8241402488571 1.9021605819011
5.50e14 671255245533 -412495.10 1.8243593389029 1.9021605816063
6.00e14 728412916123 -399030.90 1.8245582810368 1.9021605816028
6.50e14 785290803354 -219998.15 1.8247403931446 1.9021605810260
7.00e14 841912734248 -330271.47 1.8249082431040 1.9021605813540
7.50e14 898298225772 -362599.07 1.8250638545912 1.9021605814440
8.00e14 954464283498 -207253.20 1.8252088524970 1.9021605810407
8.50e14 1010426459312 -534774.22 1.8253445623449 1.9021605818329
9.00e14 1066196920739 -459168.78 1.8254720744001 1.9021605816527
9.50e14 1121787648852 -422043.55 1.8255923015512 1.9021605815759
1.00e15 1177209242304 -750443.32 1.8257060132403 1.9021605822498
1.05e15 1232470313145 -856839.63 1.8258138623433 1.9021605824537
1.10e15 1287579137984 -732612.87 1.8259164099410 1.9021605822159
1.15e15 1342543717345 -879045.52 1.8260141417697 1.9021605824791
1.20e15 1397370335220 -761338.54 1.8261074785719 1.9021605822802
1.25e15 1452065612798 -774787.26 1.8261967900949 1.9021605822983
1.30e15 1506635099560 -762644.45 1.8262824008978 1.9021605822837
1.35e15 1561084329425 -1010551.62 1.8263645987282 1.9021605826562
1.40e15 1615417411648 -785068.05 1.8264436378766 1.9021605823288
1.45e15 1669639673260 -931388.83 1.8265197475581 1.9021605825390
1.50e15 1723754585354 -761213.67 1.8265931311402 1.9021605823084
1.55e15 1777766583441 -839920.90 1.8266639732589 1.9021605824132
1.60e15 1831678961614 -851925.37 1.8267324395006 1.9021605824283
=======================================================================
4. Brun's Constant and the Error Analysis
The first order extrapolation F_2(1.6*10^15) is believed to yield
the most accurate value known to this point for Brun's constant,
(4.1)
B_2 = 1.90216 05824 +/- 0.00000 00030 .
The error estimate is believed to represent a 99 % confidence
level, although this is a subjective opinion of the author for
which definitive proof remains lacking. More precisely, the
author believes that for at least 99 % of the integers in any
sufficiently large set of distinct integers x > 1, Brun's
constant B_2 would lie between F_2(x) - E_2(x) and F_2(x) + E_2(x).
Here E_2(x) is the error bound function stated in (4.6) below,
from which the error bound in (4.1) will be obtained. The error
estimate is arrived at as follows.
First, it is assumed that the deviations F_2(x) - B_2 change sign
infinitely often; more precisely, given any x_0, however large,
there will exist integers x_1 > x_0 and x_2 > x_0 such that
F_2(x_1) < B_2 and F_2(x_2) > B_2. We will refer to
this as hypothesis [H2]. Note that although [H2] is neither
necessary nor sufficient for the Hardy-Littlewood approximation
(1.2), it will almost certainly fail (along with (4.1) in its
entirety) if Hardy-Littlewood is false. In support of [H2],
we note that if our best estimate F_2(1.6*10^15) is used for B_2,
then F_2(x) - B_2 is observed to undergo 872 sign changes over the
160081 data points recorded in the present study, with 677 sign
changes occurring beyond 10^15. Brent [2] presented a heuristic
argument for a much stronger characterization of the deviations,
one which would imply [H2], namely that the values of the scaled
deviations
(4.2)
P_2(x) = sqrt(x)*ln(x)*[F_2(x) - B_2]
(indicated in [13] by P(x)) are asymptotically normally
distributed with mean o(1) and standard deviation
sqrt(8c_2)=2.2981. A statistical analysis of P_2(x), using
F_2(1.6*10^15) for B_2 and the 160081 data points from the present
computation, indicates a distribution that is only roughly normal,
with mean -0.5294, standard deviation 0.7489, and widely distributed
sign changes. As in a similar analysis performed by the author
in [13], the standard deviation was significantly less than
Brent's prediction; however, this statistic is quite sensitive
to the value presumed for B_2.
Given [H2], we then look for the maximum computable value
of the function
(4.3)
N_2(x_1, x_2) = sqrt(x_1)*ln(x_1)*|F_2(x_1) - F_2(x_2)|
where x_1 and x_2 are integers such that x_2 > x_1 > 1.
N_2 may be considered as a "normalized" measure of the
amplitude of the oscillations in F_2(x), or alternately
as a measure of the scaled deviation of a "predictor" value
F_2(x_1) from a "terminal approximation" F_2(x_2). In contrast
to P_2, N_2 has the virtue that it is independent of the
uncertain value of B_2. If N_2 has a global maximum, or even an
upper bound, then given [H2] this must also represent an upper
bound on the absolute value of P_2, thus producing an unconditional
error bound; for any specific value |P_2(x_3)| will be exceeded by
N_2(x_3, x_4), where x_4 is chosen so that x_4 > x_3 while
F_2(x_3) - B_2 and F_2(x_4) - B_2 are of opposite sign ([H2] implies
that there is an infinite sequence of such integers x_4 for any
given integer x_3). In practice, we are unable to prove that N_2 has
a global maximum, although (1.4) would imply that P_2 is bounded.
Indeed, it is computationally impractical even to find the absolute
maximum of N_2 over all integers in the range under investigation,
1 < x_1 < x_2 <= 1.6*10^15; this would involve the
comparison of more than 10^30 data pairs (F_2(x_1), F_2(x_2)),
a computation which might bring an entire Web of teraflop computers
to its knees. What has been determined is that the absolute maximum
of N_2 over all of the (more than 10^10 such) data pairs recorded in
this study is
(4.4) N_2(1000000, 9000000) = 3.805437 .
However, additional computations reveal an even larger value of
N_2 in the vicinity of this point,
(4.5) N_2(872140, 4265159) = 4.14083 ,
where the values in both (4.4) and (4.5) have been rounded up.
Although the value 4.14083 is still not established as the sought after
upper bound on N_2 (and thus on |P_2|), the numerical evidence (zero
exceptions among more than 10^10 data pairs extending over fifteen
orders of magnitude) indicates that if any larger values of N_2 (and thus
|P_2|) exist, they must be relatively rare. Our intuitive conclusion is
that for the great majority (99 percent or more?) of integers x > 1,
it will be true that |P_2(x)| < 4.14083, producing the error bound formula
E_2(x),
(4.6)
|F_2(x) - B_2| < E_2(x) = 4.14083/(sqrt(x)*ln(x)) .
In arriving at our error estimate in (4.1), we are assuming that
(4.6) does in fact hold specifically for x = 1.6*10^15, so that
(4.7)
|F_2(1.6*10^15) - B_2| < E_2(1.6*10^15) =
4.14083/(sqrt(1.6*10^15)*ln(1.6*10^15)) ,
giving
(4.8)
|F_2(1.6*10^15) - B_2| < 0.00000 00029 57 ,
which, with rounding up for good measure, produces the error estimate
given in (4.1).
The error bound formula E_2(x) in (4.6) is similar to the one
arrived at (using different reasoning) by Brent [2]. Brent
obtains the constant 3.5 in place of 4.14083---partly because he
used an estimate for B_2 in place of F_2(x_2), partly because he
did not consider it important to perform a higher resolution search
for a larger magnitude of the scaled deviation, and partly because of
the different interpretation he placed on the resulting error bound;
Brent regarded the value 3.5 as representing approximately 1.523
of his standard deviations of P_2(x) (that is, 1.523*sqrt(8c_2)),
yielding a probability (based on the presumed normal distribution)
of approximately 88 % that B_2 would fall within the resulting error
bound. By contrast, the computed standard deviation of P_2(x) here
is only 0.7489; even given that this is artificially low due to the use
of F_2(1.6*10^15) as B_2, and taking into account the non-zero mean,
this implies that the error bound E_2(x) represents a variation of
nearly five standard deviations from the mean, additional support for
the opinion of a 99 % (or better) confidence level. On the other hand,
the error bound given by the author in [13] was specifically stated as
representing one computed standard deviation at 10^14, so it is not
especially surprising that the present value for B_2 exceeds the
previous estimate by slightly more than two standard deviations
(as computed at 10^14).
It should be noted that the scaling factor sqrt(x)*ln(x) was chosen
for two reasons---firstly, the persuasiveness of Brent's argument in
[2]; secondly, analysis of the data shows that the resulting mean
absolute value of P_2(x) remains of the same order of magnitude
(typically a few tenths) over most intervals of 10^12 from 0 through
1.6*10^15, as would be expected with a valid scaling factor.
However, other scaling factors could be employed, and the author in
fact investigated several alternatives. No other yielded superior
results; either the resulting error bound was inferior, or a
"near-maximum" value of the corresponding function N_2 (meaning one
sufficiently large to provide an error bound of equal or better
confidence level) could not be located without greatly escalating
the computational effort.
In summary, the error bound E_2(x) set forth here is more
conservative than that of Brent [2] and much more conservative than
that given in the author's previous work [13]. More than ten
thousand million pairs of first order extrapolations have agreed with
each other within this error bound, without exception. Nonetheless,
it carries no unconditional guarantee; overwhelming numerical
evidence is occasionally a subtle trap, as the conjecture
Li(x) > pi(x) so memorably illustrated. However, it
would appear that a more rigorous error bound must await a deeper
understanding of the subtleties in the distribution of the twin
primes---perhaps the discovery of a "Riemann zeta function" for
the twins.
5. Acknowledgments
The author was greatly aided by Eberhard Mattes' outstanding freeware
package emTeX [12], which includes an extended DOS implementation of
the LaTeX2e software system for composing and typesetting documents
such as this one.
References
- Jan Bohman, "Some computational results regarding
the prime numbers below 2,000,000,000," Nordisk Tidskr.
Informationsbehandling (BIT) 13 (1973), 242-244; errata, ibid.
14 (1974), 127. MR 48 #217.
- Richard P. Brent, "Irregularities in the distribution
of primes and twin primes," Math. Comp. 29:129 (1975),
43-56, MR 51 #5522. Corrigendum ibid. 30:133 (1976), 198,
MR 53 #302. Addendum reviewed ibid. 30 (1976), 379.
- Viggo Brun, "La série 1/5 + 1/7 + 1/11 + 1/13 +
1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ...,
où les dénominateurs sont 'nombres premieres
jumeaux' est convergente ou finie," (French) Bulletin
des sciences mathématiques 43 (1919), 100-104, 124-128.
- Carl-Erik Fröberg, "On the sum of inverses
of primes and twin primes," Nordisk Tidskr.
Informationsbehandling (BIT) 1 (1961), 15-20.
- Patrick Fry, Jeffrey Nesheiwat, and Boleslaw Szymanski,
unpublished work, e-mail communications (April-June, 1998).
- Patrick Fry, Jeffrey Nesheiwat, and Boleslaw Szymanski,
unpublished work at (November, 2000)
http://www.cs.rpi.edu/research/twinp/.
- J. W. L. Glaisher, "An enumeration of prime-pairs,"
Messenger of Mathematics 8 (1878), 28-33.
- F. J. Gruenberger and G. Armerding, "Statistics on the first
six million primes," UMT 73, Math. Comp. 19 (1965), 503-505.
- G. H. Hardy and J. E. Littlewood, "Some problems of 'Partitio
Numerorum' III: On the expression of a number as a sum of
primes," Acta Math. 44 (1923), 1-70. See also "Collected
papers of G. H. Hardy," Clarendon Press, Oxford, 1966, Vol. I,
561-630.
- Martin Kutrib and Jörg Richstein, "Primzahlen: Zwillinge
aus dem Parallelrechner," (German) Spektrum der Wissenschaft
(February, 1996), 26-31. See also (7 October 2000)
http://www.informatik.uni-giessen.de/staff/richstein/tab.html.
- Derrick Henry Lehmer, "Tables concerning the distribution
of primes up to 37 millions," 1957. Copy deposited in the
UMT file and reviewed in MTAC 13 (1959), 56-57.
- Eberhard Mattes, emTeX package available on the TeX Users Group
(TUG) "TeX Live 4"
CD (March, 1999), or at (July, 1999)
http://www.tex.ac.uk/tex-archive/systems/msdos/.
- Thomas R. Nicely, "Enumeration to 10^14 of the twin primes
and Brun's constant," Virginia Journal of Science 46:3
(Fall, 1995), 195-204. MR1401560 (97e:11014). Electronic reprint
available at
http://www.trnicely.net/twins/twins.html.
- Thomas R. Nicely, "New maximal prime gaps and first
occurrences," Math. Comp. 68:227 (July, 1999),
1311-1315. MR1627813 (99i:11004). Electronic reprint available at
http://www.trnicely.net/gaps/gaps.html.
- Thomas R. Nicely and Bertil Nyman, "First occurrence
of a prime gap of 1000 or greater," unpublished (21 May 1999),
available electronically at
http://www.trnicely.net/gaps/gaps2.html.
- Hans Riesel, "Prime numbers and computer methods for
factorization," 2nd ed., Birkhäuser Boston, 1994.
464 pp. + xvi. ISBN 0-8176-3743-5. MR 95h:11142.
- Pascal Sebah, unpublished work, e-mail communications (7 June 1999,
25 August 1999). Sebah's computations are to be available at
http://numbers.computation.free.fr.
- Ernst S. Selmer, "A special summation method in the theory of
prime numbers and its application to 'Brun's Sum', (Norwegian)
" Nordisk Mat. Tidskr. 24 (1942), 74-81. MR 8,316g.
- C. R. Sexton, "Counts of twin primes less than
100000," MTAC 8 (1954), 47-49.
- Daniel Shanks and John W. Wrench, Jr., "Brun's constant,"
Math. Comp. 28:125 (January, 1974), 293-299; corrigenda ibid. 28
(1974), 1183. MR 50 #4510.
- Barbra Streisand, "The mirror has two faces,"
TriStar Pictures, 1996. Director, Barbra Streisand. Screen story
and screenplay, Richard LaGravenese, based on the motion picture
"Le miroir a deux faces" (1959). Cinematographers,
Dante Spinotti and Andrzej Bartkowiak. Music, Marvin Hamlisch and
Barbra Streisand. Producers, Barbra Streisand and Arnon Milchan.
Starring Barbra Streisand, Jeff Bridges, Lauren Bacall,
Mimi Rogers, George Segal, Pierce Brosnan, Brenda Vacarro, Austin
Pendleton, and Elle Macpherson. See the first date dinner scene.
- Charles S. Sutton, "An investigation of the average
distribution of twin prime numbers," Journal of Mathematics
and Physics 16 (1937), 1-42.
- Sol Weintraub, "Four tables concerning the distribution of
primes," UMT 38, Math. Comp. 27 (1973), 676-677.
- John W. Wrench, Jr., "Evaluation of Artin's constant and the
twin-prime constant," Math. Comp. 15 (1961), 396-398.
MR 23 #A1619.
- Jörg Richstein, "Unendliche Geschwisterliebe unter Zahlen
oder Die Suche nach den Primzahlzwillingen," (German)
Überblicke Mathematik 1996/1997, 135-143, Überbl. Math.,
Vieweg, Braunschweig, 1997. MR 99f:11117.
Addendum
NOTE: This addendum was not part of the original submission for
publication.
- The latest updated count of the twin primes,
along with corresponding estimates for Brun's constant and a definitive
error bound, may be found at
http://www.trnicely.net/counts.html.
- More extensive versions of Table 1 are available at
http://www.trnicely.net/twins/tabpi2.html,
and at
http://www.trnicely.net/twins/tabpi2x.html.
- Extensive tables of the values of pi(x), determined by direct
count as part of this project, are being maintained at
http://www.trnicely.net/pi/tabpi.html,
and at
http://www.trnicely.net/pi/tabpix.html.
- The reciprocal sums S_2(x) listed in Table 1 are believed to be
correct, and correctly rounded, to all digits shown. The calculation
of these quantities was carried out in long double precision (extended
precision, 64-bit mantissa, 19 significant digits), using the Intel
FPU hardware (numeric coprocessor). However, these calculations
were also carried out to 53 decimal places in software, using the
author's modification of the BIGINT ultraprecision integer package
developed, and graciously placed in the public domain, by Arjen K.
Lenstra, Mark Riordan, and Marc Ringuette (1988-1991). The original
BIGINT package is available (19 March 2001) at
http://www.funet.fi/pub/crypt/cryptography/rpem/rpem/ (thanks to
Charles Doty for this pointer). The predecessor of this package was
Lenstra's LIP (long integer package). Its descendant is NTL (Number
Theory Library), maintained by Victor Shoup at (19 March 2001)
http://www.shoup.net/ntl/index.html.
- This note is in response to an inquiry from
Dr. Marek Wolf.
Efforts to use weighted and/or unweighted data averaging or smoothing,
or linear regression techniques, in an attempt to obtain a more
accurate value of Brun's constant, have not met with success. The
results thus obtained, at values of x < 1.6*10^15, were in some cases
slightly better, but also in some cases slightly worse, than the
values of F_2(x). "Better" or "worse" here
refers to whether or not the resulting estimate was closer to the
presumably superior estimates of B_2 obtained from more extensive
calculations, i.e., F_2(x) for a much larger value of x. However,
I have not mounted a full-scale effort using weighted or non-linear
regression; I invite third parties to carry out such an effort,
using the extensive data tables I have
provided. It may well be possible to produce a more rapidly
converging sequence of estimates by some such method; this would
amount to a successful second order extrapolation. I recommend as
critical test values for this purpose the estimates for Brun's
constant at x = 10^11 and x = 10^13; can your proposed extrapolation
produce estimates substantially superior to F_2(10^11) and F_2(10^13)?
- At the request of Professor Kenneth Hicks, Department of Physics,
Ohio University, Athens, Ohio, the following table has been
prepared.
Table 2.
Counts of primes and twin-prime pairs at primorial values of x.
==================================================================
x x pi(x) pi_2(x)
==================================================================
2# 2 1 0
3# 6 3 2
5# 30 10 5
7# 210 46 15
11# 2310 343 70
13# 30030 3248 468
17# 510510 42331 4636
19# 9699690 646029 57453
23# 223092870 12283531 896062
29# 6469693230 300369796 18463713
31# 200560490130 8028643010 425177757
37# 7420738134810 259488750744 11997649372
41# 304250263527210 9414916809095 385088898632
2*41# 608500527054420 18434013634545 738101786057
3*41# 912750790581630 27315153707442 1080390217086
4*41# 1217001054108840 36109018077717 1415982225039
5*41# 1521251317636050 44839106856053 1746723592716
6*41# 1825501581163260 ? ?
43# 13082761331670030 ? ?
==================================================================