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 0501 GMT 3 July 2009.

NOTES

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PAPERS (Unpublished)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 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.

NEW PRIME GAP OF 1476 HAS MAXIMUM KNOWN MERIT

Professor Tomás Oliveira e Silva has discovered (1 April 2009) a first known occurrence prime gap of 1476, following the prime 1425172824437699411. This gap exhibits the greatest merit (35.310308) of any gap presently known, exceeding the merit of 34.9756865 exhibited by the Herzog-Silva maximal prime gap of 1442 described above.

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 0.8447275 for Silva's gap of 1476, and 0.8483347 for the Herzog-Silva gap of 1442.

Silva's gap of 1476 is also the largest prime gap presently known below 7.2e25.

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 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 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. 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, group discussions, or unsolicited transcripts of conversations to which I was not party.

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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

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.

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