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Thomas R. Nicely's Home Page
Some Results of Computational Research in Prime Numbers
(Computational Number Theory)
Thomas R. Nicely
http://www.trnicely.net
Current e-mail address
Site last updated 0324 GMT 12 November 2008.
NOTES
Due to family illnesses, postings and responses may be delayed.
E-mail responses to the following individuals have been returned as
refused or undeliverable: Carlos Arthur Lang Lisbôa (15 May).
Please revise your acceptance list or adjust your filtering
software.
Description of research
Papers and publications
Dense prime clusters
New maximal prime gap of 1442
Baillie-PSW primality test
GNU GMP mpz_probab_prime_p pseudoprimes
Most recent counts (to 1e16) of prime
constellations and Brun's constants
Table of all known first occurrence and maximal prime gaps
Tables of first known occurrence prime gaps
Instructions for submitting prime gaps
Discovery of the Pentium FDIV flaw
Paydirt and Bowl Bound
Other works
Table of prime counts pi(x)
Tables of twin-prime counts
Tables of counts of prime quadruplets
Tables of counts of the (q, q+2, q+6)
prime triplets
Tables of counts of the (q, q+4, q+6)
prime triplets
The Morain-Alm-Andersen record certified prime gap of 337446
Skewes' problem: Li(N) <= pi(N)
Conjectures of Golomb and Dasgupta
E-mail security alert
E-mail address display policy
Downloads
Links
Proprietary marks: Disclaimer
NOTE: For simplicity, numbers of very large or very small
magnitude, appearing in some documents on this site, may be written
using the floating-point notation of FORTRAN and C. For example,
56e15 means the same thing as 56000000000000000, 5.6*10^16,
5.6·10^16, 5.6e16, 5.6·1016,
5.6×1016, etc. However, in bibliographic references,
such a number would be rendered in TeX style, thus: $5.6 \times 10^{16}$.
DESCRIPTION OF RESEARCH
Code written primarily in GNU C, and distributed asynchronously across
available personal computers running under extended DOS, Windows, and
GNU/Linux, is employed to enumerate primes, prime gaps, prime
constellations (twins, triplets, and quadruplets) and their reciprocal
sums (to extrapolate estimates for the corresponding Brun's constants).
Some related computational results obtained by other researchers
are also reported here.
PAPERS (Unpublished)
The Baillie-PSW primality test.
Includes GNU C source and executable for implementing both
the standard and strong versions of the Baillie-PSW and
Lucas-Selfridge primality tests, as well as the extra strong
Lucas test. Original posting 10 June 2005.
GNU GMP mpz_probab_prime_p pseudoprimes.
Counterexamples for the GNU GMP primality testing function.
Original posting December 2004.
"New evidence for the infinitude
of some prime constellations" (20 July 2004).
"Enumeration to $1.6 \times 10^{15}$
of the prime quadruplets" (23 August 1999).
"Enumeration to $1.6 \times 10^{15}$
of the twin primes and Brun's constant" (16 July 1999).
"First occurrence of a prime
gap of 1000 or greater," Thomas R. Nicely and Bertil Nyman
(21 May 1999). The addendum includes listings of
the largest prime gaps
(deterministic and probabilistic) found to date.
Conjectures of Golomb and Dasgupta.
Do n/pi(n) and n/pi_2(n) take on all sufficiently large positive
integer values? Original posting 6 September 2005.
PAPERS (Published)
"New prime gaps between $10^{15}$ and
$5 \times 10^{16}$," Bertil Nyman and Thomas R. Nicely,
Journal of Integer Sequences 6 (2003), Article 03.3.1, 6 pp.
(electronic). MR1997838 (2004e:11143). Published 13 August 2003.
Available in various formats (DVI, PS, PDF, LaTeX) at the home page
of the Journal
of Integer Sequences.
"A new error analysis
for Brun's constant," Virginia Journal of Science
52:1 (Spring, 2001) 45-55, MR 1853722 (2003d:11184).
"New maximal prime gaps and
first occurrences," Mathematics of Computation
68:227 (July, 1999) 1311-1315, MR 1627813 (99i:11004).
"Enumeration to $10^{14}$ of the
twin primes and Brun's constant, " Virginia Journal of
Science 46:3 (Fall, 1995) 195-204, MR 1401560 (97e:11014).
TABLES OF PRIME GAPS
A listing of all first occurrence,
maximal, and first known occurrence prime gaps of 1 to 1998,
as well as all other prime gaps exceeding 999 which lie
below 5e16.
Tables of first known occurrence prime gaps, of measures:
2000 to 3998.
4000 to 5998.
6000 to 7998.
8000 to 9998.
10000 to 14998.
15000 to 19998.
20000 to 24998.
25000 to 29998.
30000 to 34998.
35000 to 39998.
40000 to 44998.
45000 to 49998.
50000 to 59998.
60000 to 69998.
70000 to 99998.
100000 to 999998.
1000000 to 999999998.
Note that, due to bandwidth limitations, the above tables display
truncated forms of initiating primes which exceed 200 characters
in length. The file containing the complete specifications of
abbreviated primes, and all recorded first occurrence, maximal,
and first known occurrence prime gaps, exceeds 30 MB in size.
My presently available bandwidth renders it impractical to
maintain this file online and updated. However, I have made
available the zipfile merits.zip
(216K), which contains a text file specifying the measure G and
the merit M=G/ln(p_1) for all known first occurrence and first
known occurrence prime gaps. This file should be of assistance
in determining whether or not some newly discovered gap constitutes
a new first known occurrence.
The prime gap listings were last updated 0200 GMT 12 November 2008.
OTHER TABLES
A table of pi(x), the count of primes,
with related functions. Values to 1e16 at intervals of 1e12.
A table of pi_2(x), the count of
twin-prime pairs, with related functions, to 1e16 at intervals
of 1e12..
A table of pi_4(x), the count of
prime quadruplets (q, q+2, q+6, q+8), with related functions, to
1e16 at intervals of 1e12.
A table of pi_3a(x), the count
of prime triplets (q, q+2, q+6), with related functions, to 1e16
at intervals of 1e12.
A table of pi_3b(x), the count
of prime triplets (q, q+4, q+6), with related functions, to 1e16
at intervals of 1e12.
Tomás Oliveira e Silva has computed the most extensive
tables of pi(x) and pi_2(x) of which I am aware.
Chris K. Caldwell maintains at his
Prime Pages an extensive compilation of values of pi(x).
Xavier Gourdon, Pascal Sebah, and Patrick Demichel have computed
the value of pi(x) for some extremely large values of x (e.g.,
4e22).
PENTIUM FDIV FLAW
Original e-mail message
announcing the discovery of the Pentium divison flaw,
30 October 1994.
A personal FAQ regarding the Pentium
division flaw. Updated 6 February 2008.
An account by Richard M. Smith,
President of Phar Lap Software, Inc., of the spread of the Pentium
flaw announcement across the Internet during the first few days.
pentbug.zip, a zipfile containing
the C source code (pentbug.c) and corresponding DOS executable
(pentbug.exe) for a program which will check for the flaw.
"The Pentium Division Flaw," Virginia Scientists
Newsletter, V:1 (April, 1995), 3.
Untitled article concerning the Pentium division flaw,
San Francisco Examiner, 18 December 1994, p. B-5.
OTHER WORKS
Problem Proposal #1109, Mathematics Magazine 53:5 (November, 1980),
300 (with solution), "When will spring next begin on March
21st in the United States?" (Answer: 2103 A.D.)
"Calculation of the Gregorian Easter cycle," public
lecture (October, 1977). The period of Easter in the Gregorian
calendar, as presently calculated by the Roman Catholic and
Protestant churches, was shown to be 5,700,000 years. The zipfile
easter1.zip contains GNU C source
code and a DOS/Wintel executable for calculating the dates of
Easter Sunday.
"Special techniques for the solution of a singular
integral equation," doctoral dissertation, applied
mathematics, University of Virginia, Charlottesville, 1971.
Advisor: Gordon E. Latta.
"Electronic structure of open-shell doublet-state molecules:
application to CN," master's thesis, theoretical physics,
West Virginia University, Morgantown, 1965. Advisor: Harvey
N. Rexroad.
The PAYDIRT and BOWL BOUND football simulation board games
(see below).
See Downloads for free software.
PAYDIRT AND BOWL BOUND
The following information is provided in response to numerous
inquiries.
For most of the period from 1977 to 1995, I carried out design and
development for the football simulation board games Paydirt (pro)
and Bowl Bound (college), produced and distributed commercially
by Avalon Hill Game Company (Baltimore, Maryland) and Sports
Illustrated Enterprises. Commercial support of these games was
suspended in April, 1995, and I retired from development in
February, 1996. Avalon Hill Game Company was later acquired by
Hasbro, Inc., and commercial design, production, and distribution
of both games was suspended indefinitely. It appears that Hasbro
retains the rights to both games at this time.
Inquiries regarding these games and their team charts should
be directed to Mr. Matt Floray, who has undertaken design,
revision, production, and distribution in the interim. Mr. Floray
has been in contact with Hasbro, Inc., regarding efforts to bring
the games back onto the market. Mr. Floray can be contacted at
butchcassidy(AT)earthlink(DOT)net; at
sundancekid63(AT)sbcglobal(DOT)net; or at 213-576-3238.
Mr. Floray has access to all the data files, documentation,
algorithms, and computer codes that I used to design Paydirt
and Bowl Bound charts from 1977 to 1995, and hopes to
produce both new and revised charts for these games.
Incidentally, the 1984, 1985, 1986, and 1987 Paydirt team charts
were not my work...despite the fact that my name appears
(unauthorized) on many of them.
NEW MAXIMAL PRIME GAP OF 1442
Professor
Tomás Oliveira e Silva and
Professor Siegfried
"Zig" Herzog of Penn State University (Mont Alto), using
computer codes written by Silva, have completed (26 April 2007) an
exhaustive scan of all prime gaps through 1e18, as part of the
process of confirming Goldbach's conjecture for all n < 1e18.
Portions of this interval had been previously scanned for prime gaps
by other researchers.
As a result, a first known occurrence prime gap previously discovered
(21 November 2005) by Herzog---the gap of 1442 following the prime
804212830686677669---is now confirmed as a first occurrence and
maximal prime gap, the largest one presently known. The Herzog-Silva
maximal gap of 1442 succeeds the previous record maximal prime gap of 1370
following the prime 418032645936712127, discovered 10 September 2006
by Professor Donald
E. Knuth of Stanford University. |
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