The largest value of x for which π(x) has been computed exactly is
π(10^25) = 176846309399143769411680
a result announced 01 June 2013 by J. Buethe, Jens Franke, A. Jost,
and T. Kleinjung, and relayed to me by Neil J. A. Sloane, formerly of
A.T. & T. Bell Labs (thanks also to Bill McEachen for reminding me
this needed to be updated). The technique was an analytic method based
on Weil's explicit formula. The result is independent of the Riemann
PENTIUM FDIV FLAW
- A personal FAQ regarding the Pentium
division flaw. Bibliography attached. Last updated
0900 GMT 19 August 2011.
- Original e-mail message
announcing the discovery of the Pentium division flaw,
30 October 1994.
- 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 executables
(pentbug.exe and bug16bit.exe) for a program which will check for
- The Pentium division flaw. Thomas R. Nicely. Virginia Scientists
Newsletter, Volume 1 (April, 1995), p. 3.
- Untitled article concerning the Pentium division flaw. Thomas R.
Nicely. San Francisco Examiner (18 December 1994), p. B-5.
- 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
- "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
- The PAYDIRT and BOWL BOUND football simulation board games
- See Downloads for free software.
PAYDIRT AND BOWL BOUND
The following information is provided in response to numerous
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
<firstname.lastname@example.org>; 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.
Transcripts of these charts may also be available from various
third parties. I do not authorize, forbid, or restrict sales or
distribution by such parties, known or unknown. Since I am not a
participant or stakeholder in such operations, I do not accept
legal responsibilty or liability for such sales or products. I do
ask that my copyright notice be retained on each chart.
Incidentally, the 1984, 1985, 1986, and 1987 Paydirt team charts
(as shipped by Avalon Hill) were not my work...despite the fact
that my name appears (unauthorized) on many of them.
NEW PRIME GAP OF MAXIMUM KNOWN MERIT
Michiel Jansen has discovered (03 January 2012) a new first known
occurrence prime gap of measure G=66520 following the 816-digit prime
P1=1931*1933#/7230 - 30244 (where 1933# indicates the product of all
primes from 2 to 1933 inclusive). The merit M=G/ln(P1) of this gap is
M=35.4244594, the largest merit of any known prime gap. The endpoints
of the gap have been certified as primes deterministically, using
Marcel Martin's TITANIX code
(version 2.1.0 from 2001; since evolved into PRIMO).
However, Bertil Nyman's maximal gap of 1132, following the prime
1693182318746371 (discovered 24 January 1999), continues to exhibit
the greatest known value (0.9206386) of the
Cramér-Shanks-Granville ratio G/ln²(p_1); this ratio is
0.0188648876 for Jansen's gap, and 0.8447275 for Silva's maximal gap
of 1476 (see below). The limit superior of this ratio has been
conjectured to be one (or some even larger value); see the discussion
in "New prime gaps between 1e15 and
NEW MAXIMAL PRIME GAP OF 1476
Tomás Oliveira e Silva (Universidade de Aveiro, Portugal) and
"Zig" Herzog (Penn State University, Mont Alto), using
computer codes written by Silva, have completed (24 July 2009) an
exhaustive scan of all prime gaps through 1500e15, as part of the
process of confirming Goldbach's conjecture for all n to this bound.
Portions of this interval had been previously scanned for prime gaps
by other researchers. The
upper bound of exhaustive scan has since been extended (04 April 2012)
to 4000e15 by Silva, Herzog, and Silvio Pardi.
As a result, a first known occurrence prime gap previously discovered
(01 April 2009) by Silva---the gap of 1476 following the prime
1425172824437699411---is now confirmed as a first occurrence and
maximal prime gap, the largest one presently known. Silva's maximal gap
of 1476 succeeds Herzog's previous record maximal prime gap of 1442
following the prime 804212830686677669 (discovered 21 November 2005).
E-MAIL SECURITY ALERT
My current e-mail address is always available
elsewhere on this site.
If you receive an e-mail claiming to be from my address (or some
slight variation of my address), which is threatening, abusive,
solicitous, commercially oriented, questionable in nature, or otherwise
suspicious, treat it as a fraudulent act of vandalism on the part of
some third party; ignore its contents and delete it! I DID NOT
Be aware that malicious parties and spammers frequently spoof legitimate
e-mail addresses, including my own, using forged headers. My own
e-mails will always have distinctive identification headers, aside from
those inserted by the mail provider. On the rare occasions when I send
attachments with e-mails, it will be with the prior permission of the
recipient, or there will be a clear explanation within the message of
the contents of the attachment. Furthermore, I never include
Active-X controls in e-mail (although the e-mail providers, such as
Hotmail, might add such features without my permission, just as they
append commercial footers without warning).
If possible, send your e-mail messages as plain text; avoid
HTML and rich text, especially in e-mails containing data to be
processed. Attachments and large data files should be sent as zipfiles
(this protects the contents from corruption by the mailers). Please DO
NOT send embedded images (jpg, gif, bmp, etc.) in your messages,
as these constitute a security hole for viruses and worms, and create
a serious bottleneck in e-mail processing. If such images are deemed
critical, send them in separate zipped attachments.
I have provided
for submitting lists of prime gaps.
Make sure that your subject line is to the point---otherwise, your
message might be deleted, unread, as likely spam. Also, if you are
seeking information or advice, please send, on your own behalf, a clear
and concise explanation of the question or problem. Ordinarily, I will
not reply to carbon copies, inquiries by a third person on behalf of
others, or unsolicited transcripts of conversations, dialogues, or
group discussions to which I was not party.
If your zipfiles or other attachments are extremely large
(over 10MB), I do not advise sending them via e-mail. For such
extremely large files, provide instead a pointer to a website from
which I can download the file.
- NOTE: These applications are distributed as freeware,
copyright © 2013 Thomas R. Nicely, released into the public
domain by the author, who disclaims any legal liability arising from
their use. All are 32-bit console (terminal, non-GUI, tty,
command-line, shell) applications, optimized for a window size of
80x25 or greater. Unless otherwise stated, any source code provided
is in GNU C (4.5.2 or later),
including the GMP 5.0.2 and
MPFR 3.0.1 libraries (earlier
versions of these packages may or may not require modifications
of the source codes). Primary development and testing are carried
out in the 32-bit Windows x86 environment, on 32-bit standalone
machines with administrative privileges, using the
MinGW/MSYS compilers and
development environment (version 28 January 2011). Any executables
provided are native to 32-bit Windows (98SE and later). However,
efforts are being made to maintain portability to other compilers
and platforms, including GNU/Linux (SUSE 10.x as root user), Cygwin,
Digital Mars, DJGPP, and Borland (version 5.51). Compatibility with
these compilers and platforms is in some cases limited by their lack
of support for GNU extensions, C99, C0x, GMP, MPFR, the long double
data type, various glibc functions, the conio functions of
DOS/DJGPP/Borland, and by non-standard interfaces to 64-bit integers.
Support for compilers and platforms other than MinGW/MSYS/Windows
is at an early beta stage, and will be extended in breadth and depth
as time and resources allow. Support for MSVC, Cygwin, and 64-bit
compilers is only at alpha level.
Compilation of the sources will typically require a command line
gcc xxx.c trn.c conio3.c -std=gnu99 -lm -lmpfr -lgmp -oxxx.exe
where xxx.c is the name of the main source file; the exact
command line parameters will depend upon your operating environment
and the specific code being compiled. The support library trn.c
(and its header trn.h), and the GMP library (4.3.1), will be needed
for the great majority of the codes; MPFR (2.4.1) will be required
for some applications; while the support library conio3.c (and its
header conio3.h) will only be required if the code calls conio
console functions (such as gotoxy, wherex, etc.) and is being
compiled outside of DJGPP and Borland C. No makefiles are required.
The extensions of the gnu99 standard (including most of C99) are
used throughout these codes.
If you are *not* linking with GMP or MPFR, you will need to append
the qualifiers -D__NOGMP__ and/or -D__NOMPFR__
on the command line.
Note that, in general, if you wish to recompile the codes or examine
the source code of the support routines, you will need to download
trn.zip separately in order to obtain the
files trn.c and trn.h (a few packages still include older,
dedicated versions of these support files).
- trn.zip, a zipfile (61K) containing the
latest revisions of the source code (trn.c) and header file (trn.h)
for the support routines called by many of the downloadable applications
listed below (some of the applications include their own support files,
or are self-contained). Multiple platforms.
Last updated 1000 GMT 26 January 2012.
- pix4.zip, a zipfile (58K) containing the
source code and a Wintel executable for calculating π(x) using the
Legendre-Meissel-Lehmer algorithm. Command-line syntax: pix4 [LB] UB.
Run time for x near 1e9 is less than one second; x may be as large as
1e19, but execution time balloons to several hours near 1e15 or
1e16, and memory requirements also increase. The LML algorithm is
written as a function (sllLML) in the module lml.c, which may be
called from your own code by recompilation and linking. For
recompilation, you will also need to download the library files
trn.c and trn.h in trn.zip, and include
the command-line qualifier -std=gnu99. Not compatible with 64-bit
compilers or operating systems; does not require or use GMP or MPFR.
Last updated 0545 GMT 20 July 2011.
- conio3.zip, a zipfile (9K) containing the
latest revisions of the source code (conio3.c) and header file
(conio3.h) for a library of functions which emulate some of the
conio functions (gotoxy, wherex, etc.) native to DJGPP and Borland C
in DOS console environments. Needed only if the main code calls such
functions and is being compiled outside of DJGPP and Borland C.
Portions of this code, notably the Win32 sections, were adapted from
the package devpak CONIO 2.0 (CONIO2), written and released to the
public domain by Hongli Lai, tkorrovi, Andrew Westcott, and Michal
Molhanec, and targeted at the Win32 MinGW/Dev-C++ platform. The
original CONIO 2.0 is available
here; thanks to David Hoke for this pointer, and for his own
adaptation of CONIO 2.0. Multiple platforms (but does not support
Unicode/wchar_t). Last updated 1000 GMT 26 January 2012.
- bpsw1.zip, a zipfile (229K)
containing the source code and executable for an application which
illustrates the standard and strong versions of the
Baillie-PSW primality test, as well as
the standard and strong Lucas-Selfridge tests and the extra strong
Lucas test. Requires trn.zip and GMP,
but does not require MPFR (recompile with -D__NOMPFR__).
The actual code implementing most of these tests is contained in
the support module trn.c. Last updated 0200 GMT 28 March 2013.
- cglp4.zip, a zipfile (242K)
containing the source code and executable (MinGW/Win32) for an
application which checks prime gaps for validity, using the
strong Baillie-PSW primality test.
Requires GMP, trn.zip, and possibly conio3.zip. Multiple platforms.
Last updated 1200 GMT 13 January 2013.
- easter1.zip, a zipfile (39K)
containing source code and an executable for calculating the date
of Easter Sunday for specified years. Support is provided for
both the Western Church (Catholic/Protestant) and Eastern Orthodox
algorithms, and for both the Gregorian and Julian (Old Style)
calendars. No warranty expressed or implied; this code has not
been endorsed or approved by any religious institution,
organization, or authority. Last updated 0950 GMT 23 April 2010.
- factor1.zip, a zipfile (190K)
containing source files (GNU C with GMP) and an executable for a
code which illustrates some algorithms used for factoring integers,
including small prime generation, trial divisors, Brent's variation
of Pollard's rho method, Pollard's (p-1) method, and a partial
implementation of the ECM method. Expression parsing capability is
included to allow input in formula form, such as factor1
"2**150 + 1" (command line arguments may require enclosure
in double quotes under operating systems such as Windows XP). No
claim is made that this code is "state of the art" or
"research caliber"; it is most certainly no threat to
current encryption schemes. It may eventually be improved by
incorporating additional factoring algorithms. Last updated
0900 GMT 08 July 2011.
- lirz.zip, a zipfile (252K) containing source,
documentation, data files, and an executable for the purpose
of computing the number-theoretic functions Li (logarithmic integral);
HL2, HL3, and HL4 (Hardy-Littlewood integral approximations); and R(x),
Riemann's prime number function/formula. Routines are included for
GNU C with ultraprecision (GCC 4.1.2, GMP 4.2.1, MPFR 2.2.1), GNU C
with long double precision, UBASIC 8.8f (ultraprecision), and
Mathematica 2.1 (ultraprecision).
Last updated 0530 GMT 03 October 2008.
- pentbug.zip, a zipfile (55K)
containing the C source code (pentbug.c) and executables (pentbug.exe
and bug16bit.exe) for an application which will check for the Pentium
FDIV flaw. The executable bug16bit.exe is a 16-bit executable intended
for use in standalone DOS, such as systems prior to Windows 95 (which
might appear on machines containing flawed processors). Last updated
0700 GMT 20 August 2011.
- pi2.zip, a zipfile (148K) containing
the C source codes (pi2e.c and pi2f.c) and executables (pi2e.exe
and pi2f.exe) for programs illustrating some practical techniques
for generating the twin primes and tabulating their properties.
The pi2f code takes advantage of the sieve of Eratosthenes; the
pi2e code uses the simple square-root test for primality. The
pi2f code is faster in most cases, but either one can enumerate
all the twin primes below 1e6 in less than one second on a
600 MHz Celeron; pi2f can enumerate all those below 1e8 in under
15 seconds. Last updated 2100 GMT 22 November 2004.
- pix.zip, a zipfile (176K) containing the
C source codes and executables for enumerating the primes and π(x).
Three algorithms are illustrated, using the GMP mpz_probab_prime_p
and Baillie-PSW tests, trial divisors to the square root, and the
sieve of Eratosthenes over byte arrays. Last updated
0530 GMT 15 July 2009.
- td2k.zip, a zipfile (20K) containing
the source code (td2k.ub) and documentation (td2k.txt) for a
UBASIC application designed
for discovering new first known occurrence prime gaps. This is a
fully operational research production code. If you download and
use it, I encourage you to notify me of any new first known
occurrence prime gaps you discover; I will then post them (with
proper attribution and credit) in my lists. NOTE: The input and data
files of td2k are incompatible with those of the previous
version, td2j. Runs begun with td2j should be completed with
td2j, or re-started from scratch with td2k.
Last updated 0225 GMT 29 April 2005.
- UBASIC (725K), a freeware
GW-BASIC-like interpreted programming environment developed by
Professor of Mathematics Yûji Kida of Rikkyo University,
Japan. UBASIC features easily accessible ultraprecision integer
and floating point arithmetic (hundreds of digits), as well as
numerous additional intrinsic functions of specific interest in
computational number theory. No computational number theorist
should be without UBASIC! Also very effective for classroom
instructional use. The zipfile
provided here contains Version 8.8f (08 October 2000), the
last stable version of which I am aware. See also
- WARNING: Be aware that, due to the peculiar command-line
parsing algorithm incorporated in recent versions of Microsoft
Windows, mathematical expressions in command lines should, to
avoid misinterpretation, be specified within double quotes;
mycode "2**150 + 1"
This syntax is also valid under DOS and older versions of
Windows, but the double quotes were optional in those
operating environments. Depending on the programming
language, it may also be necessary (within the source code)
to strip off the double quotes and/or concatenate command-line
arguments. Finally, replacing the exponentiation operator "^"
(a particularly troublesome token for Windows) with "**" (as
in FORTRAN/COBOL) may be helpful, if the application permits.
Following are some websites of relevance to mathematics in general,
and number theory in particular. Note that these pages may open in a
new browser window.
DISCLAIMER: No endorsement of, or by these sites is
expressed or implied, and Thomas R. Nicely accepts no responsibility
or liability in consequence of their access or content. Furthermore,
no endorsement, expressed or implied, is granted to other sites
which link to this site (with or without my authorization), and no
responsibility or liability is accepted for the access, content,
accuracy, or integrity of any external site.
- Complete counts and reciprocal sums
of the prime constellations from Nicely's computations (1993-2009), as
maintained at the Department of Mathematics, University of Washington
(Seattle). NOTE: These data files are quite large (over 60MB each,
even for the zipped versions), including more than two million data
points from 0 to 2e16.
- The GNU project ("GNU's Not UNIX"),
launched in 1984 to develop and provide as free software
(under the terms of the
GNU GPL, Lesser
GPL, and FDL licenses) a complete UNIX-like operating system,
including utilities, applications, and development tools. Linux is
one kernel for the GNU operating system. Supported by the
Free Software Foundation.
- The GMP (GNU MP) multiple
precision software package. Excellent for ultraprecision integer
arithmetic; incomplete support for floating-point arithmetic and
DOS/Windows platforms. Version 4.2.1 or later recommended.
- MPFR, a C library for
multiple-precision floating-point computations with correct rounding,
reliable precision control, and compatibility with the ANSI/IEEE
754-1985 standard. MPFR is based on (and assumes pre-installation of)
the GMP multiple-precision library. It is open-source software,
distributed under the terms of the GNU Lesser GPL license. MPFR is
supported and maintained by French teams at
LIP. It provides many features
unavailable with the GMP mpf_t data type and libraries, notably a
large collection of transcendental functions. Version 2.2.1 or later
- MPC, a portable library
written in C for arbitrary precision arithmetic on complex numbers
providing correct rounding. Ultimately, it should implement a
multiprecision equivalent of the C99 standard. MPC builds upon the
GNU MP and the GNU MPFR libraries. Written and maintained at
INRIA by Andreas Enge, Philippe
Théveny, and Paul Zimmermann. Distributed under the GNU LGPL
as free software.
- MinGW, minimalist GNU for Windows.
MinGW is a collection of freely available and freely distributable
Windows specific header files and import libraries, combined with GNU
toolsets that allow one to produce native Windows programs which do
not rely on any third-party C runtime DLLs. MinGW is distributed in
conjunction with MSYS, a Minimal SYStem (shell) providing POSIX/Bourne
configure, make, and libtool services within 32-bit Windows. MinGW
and MSYS together provide a scalable development environment for GCC
applications within 32-bit Windows, with support for GMP and MPFR.
The executables require no third-party DLLs, but are specific to the
Win32 platform, and rely on the presence (and share some of the
shortcomings) of certain Microsoft system DLLs (e.g., MSVCRT.DLL).
The deficiencies of MinGW with regard to long doubles, 64-bit integers,
and conio are partially remedied by the functions incorporated in the
trn and conio3 libraries. Further
comments are provided.
- DJ Delorie's DJGPP port
of the GNU GCC compilers and utilities (including GMP) to the
- Home of the C
standard, currently C99 (ISO/IEC 9899:1999), as revised in
2001, 2004, and 2007. Work is underway on the succeeding standard,
- Home of the C++
standard. The current standard is ISO/IEC 14882:2003 (C++03),
the union of the C++98 standard ISO/IEC 14882:1998 and the 2003
corrigendum. Work is underway on the proposed C++0x standard.
- Tomás Oliveira
e Silva's projects in computational number theory.
- The home page of
E. Knuth of Stanford University.
- Jens Kruse Andersen's site featuring
The Top-20 Prime Gaps, the successor to a compilation
maintained prior to February 2004 by Paul Leyland.
- The Prime
Pages, Chris K. Caldwell, University of Tennessee at Martin.
Includes an elementary introduction to prime numbers and number
- The Number Theory Web,
maintained by Keith Matthews, University of Queensland, Brisbane,
- MathWorld, a Wolfram Web
resource, maintained by Eric W. Weisstein.
Mathematical constants and computations. Ultraprecision mathematical
constants; very fast and very compact algorithms and codes for the
evaluation of certain classical mathematical constants; evaluation of
π(x) for extremely large x ( > 1e20). Site maintained by Xavier
Gourdon and Pascal Sebah. Sebah also plans to post at this site
periodically updated results of his own enumeration of the twin
primes and the associated estimates of Brun's constant.
- The Mathematics WWW Virtual
Library of Florida State University.
- The Penn State index of
Mathematics Websites around the world.
- The American Mathematical Society (AMS).
- The Mathematical Association of
- The Society for Industrial and Applied
- The Society of Actuaries (SOA).
- The Association for Computing Machinery
- PARI-GP, a software
package for computer-aided number theory, including the ultraprecision
libpari C libraries and the gp programmable interactive calculator.
Targeted at UNIX platforms, with some DOS/Wintel support. Site
maintained by Henri Cohen and Karim Belabas.
- GIMPS, the Great Internet
Mersenne Prime Search, a group enterprise using distributed computing
across the Internet to search for new primes of the form 2^p - 1,
where p is prime. The Prime95 code employed uses George Woltman's
gwnum library, highly optimized for x86 processors, as well as
code by Richard Crandall.
- PrimeForm/GW, highly
optimized x86 software by Yves Gallot and George Woltman, designed
to perform compositeness tests, probabilistic primality tests, and
(limited) deterministic primality tests. This software
incorporates Woltman's gwnum library, the core of the Prime95 code
used by GIMPS.
- TtH, Ian
Hutchinson's TeX to HTML translator.
- UPX, "the Ultimate
Packer for eXecutables". UPX is a free, portable, extendable,
high-performance executable packer for several different executable
formats. It achieves an excellent compression ratio and offers very
fast decompression. Executables suffer little or no memory overhead
or other drawbacks for most of the formats supported, because of
in-place decompression. UPX is copyrighted software distributed under
the terms of the GNU General Public License, with special exceptions
granting free usage for commercial programs as stated in the UPX
License Agreement. Maintained and copyrighted by Markus F. X. J.
Oberhumer, László Molnar, and John F. Reiser (all rights
- DOSBox, an emulator that recreates
an MS-DOS compatible environment (complete with sound, input, graphics
and even basic networking). This environment is accurate enough to run
many classic MS-DOS games completely unmodified. DOSBox has been ported
to many different platforms, including Windows, BeOS, Linux, and Mac
OS X. I can personally testify that DOSBox allows me to run Derive XM
3.01, Scrabble Deluxe 1.0 (29 April 1991), and Chess88 (version 2.0,
16 March 1984) in full-screen mode under Vista SP1. DOSBox
is free of charge and open-source, published under the
GNU GPL license.
Copyright DOSBox Team.
- Spybot - Search & Destroy, a
freeware application designed to detect and remove spyware of
different kinds from your computer. Spybot provides a free software
alternative to costly proprietary anti-spyware programs. Also, Spybot
is a passive (manual or on-demand) anti-spyware application, and thus
avoids the python-like grip of some commercial anti-spyware packages,
whose on-access real-time scanners can seriously impact the performance
and interface of a system. Frequent signature updates are made
available. Copyright Safer Networking Ltd., County Wicklow, Ireland.
- ClamWin, a free anti-virus
application for various platforms. ClamWin provides a free software
alternative to costly proprietary anti-virus programs. Also, ClamWin
is a passive (manual or on-demand) anti-virus application, and thus
avoids the python-like grip of commercial anti-virus packages, whose
on-access real-time scanners can seriously impact the performance
and interface of a system. Based on the
Clam AntiVirus engine, ClamWin
is an open source code released under the terms of the
Public License. Daily virus signature updates are provided.
Copyright ClamWin Pty Ltd.
PROPRIETARY MARKS: DISCLAIMER
Any words, symbols, abbreviations, phrases, marks, or other tokens
which appear on this site, and are trademarked, copyrighted, or
otherwise considered the legal property of corporate, governmental,
academic, or private entities, are recognized as being by law the
property of their respective legal owners. The author of this site
has no commercial association with any of these entities, or with
their representatives, products, or vendors, and the information
and opinions on this site are not to be construed as reflecting the
endorsement, position, opinion, approval, or participation of any
of these entities, or of their representatives or vendors. It
remains the personal opinion of the author that current laws
regarding "intellectual property rights" are oppressive
of free speech, impede the spread of knowledge, and are contrary to
the public interest.
Dates and times on this site are either Greenwich Mean
Time (GMT, UTC, Zulu) or USA Eastern Time (EST=GMT-5 or EDT=GMT-4),
Freeware copyright © 2014 Dr. Thomas R. Nicely, Professor Emeritus
of Mathematics, Lynchburg College, 1501 Lakeside Drive, Lynchburg VA
24501-3113 USA <http://www.trnicely.net>. Released into the public
domain by the author, who disclaims any legal liability arising
from its use.