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Some Results of Computational Research in Prime Numbers
(Computational Number Theory)

Thomas R. Nicely

http://www.trnicely.net
Current e-mail address

NOTE: Due to family illness, postings and responses may be delayed.
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PAPERS (Unpublished)

PAPERS (Published)

TABLES OF PRIME GAPS

OTHER TABLES

PENTIUM FDIV FLAW

OTHER WORKS

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 Tom�s 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. In addition to being the largest presently known maximal prime gap, the Herzog-Silva gap has the greatest merit (34.9756865) of any known gap, exceeding the merit of 32.28254764 exhibited by Bertil Nyman's maximal gap of 1132 following the prime 1693182318746371 (discovered 24 January 1999). However, Nyman's 1132 gap continues to exhibit the greatest known value (0.9206386) of the Cram�r-Shanks-Granville ratio G/ln�(p_1); this ratio is 0.8483347 for the Herzog-Silva gap.

The Herzog-Silva gap is also the largest prime gap presently known below 6e25.

See the page First occurrence prime gaps for a complete listing of all known first occurrence and maximal prime gaps; also, the addendum to the paper New maximal prime gaps and first occurrences for a further discussion of the new maximal prime gaps.

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 SEND IT!

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 links, embedded images, JavaScript, VBScript, or Active-X controls in e-mail (although the mail providers, such as Hotmail, might add such features without my permission).

If possible, send your e-mail messages as plain text. 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 detailed instructions 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.

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.

E-MAIL ADDRESSES MASKED

As a general policy, literal e-mail addresses are no longer published on this site. A few documents have been left unaltered, due to possible historical relevance, in which literal e-mail addresses appear, but it is unlikely (after nearly a decade) that these addresses remain valid.

This is part of an effort (probably futile) to hinder the trillions of agencies ceaselessly scouring the Web for e-mail addresses, collecting them for spamming or abusive purposes. This is also the principal reason for the lack of any direct e-mail link to the author.

DOWNLOADS 

gcc xxx.c trn.c conio3.c -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 (4.2.1+), will be needed for the great majority of the codes; MPFR (2.2.1+) 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.

  • trn.zip, a zipfile (44K) 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 0500 EST 1 March 2008.
  • 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 outeside 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 0530 EDT 22 August 2007.
  • bpsw1.zip, a zipfile (187K) containing the source code, support files, and executable for implementing 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. GNU/Linux compatible. Last updated 2350 EST 9 February 2007.
  • cglp4.zip, a zipfile (133K) 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 0430 EDT 31 August 2007.
  • easter1.zip, a zipfile (57K) 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 0445 EST 21 December 2005.
  • factor1.zip, a zipfile (131K) 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. An expression parser 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 1300 GMT 26 January 2005.
  • lirz.zip, a zipfile (83K) containing source, documentation, data files, and an executable for the purpose of computing the number-theoretic functions Li (logarithmic integral); L2, L3, and L4 (Hardy-Littlewood integral approximations); and R(x), Riemann's prime number function/formula. Routines are included for GNU C (GCC 3.04, long double precision), UBASIC 8.8f (ultraprecision), and Mathematica 2.1 (ultraprecision). Last updated 0500 GMT 25 April 2004.
  • pentbug.zip, a zipfile (55K) containing the C source code (pentbug.c) and executable (pentbug.exe) for an application which will check for the Pentium FDIV flaw. Last updated 26 April 2003.
  • 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 (209K) containing the C source codes and executables for enumerating the primes and pi(x). Three algorithms are illustrated, using the GMP mpz_probab_prime_p function, trial divisors to the square root, and the sieve of Eratosthenes over byte arrays. Last updated 0100 GMT 29 December 2004.
  • 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 Y�ji Kida of Rikkyo University, Japan (ftp://rkmath.rikkyo.ac.jp/pub/ubibm/). 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 (7 October 2000); see also ftp://rkmath.rikkyo.ac.jp/pub/ubibm/.
  • 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; e.g.,

    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.

    • LINKS

    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 content or access of any external site.

    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 to free speech and contrary to the public interest.

    NOTICE:� I have not been affiliated with Lynchburg College since 6 July 2000.

    Copyright � 2008 Thomas R. Nicely. All rights reserved. This document and others on this site may be reproduced and distributed for educational and non-profit purposes. No warranties are expressed or implied for the content on this site. Unless otherwise noted, all dates and times on this site are USA Eastern Time (EST=GMT-5 or EDT=GMT-4).

    Site last updated 2310 EDT 19 March 2008.