Some Results of Research in Computational Number Theory

Dr. Thomas R. Nicely

University of Lynchburg Department of Mathematics

NOTES: 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 some bibliographic references, such a number may be rendered in TeX style, thus: $5.6 \times 10^{16}$. Also, the default on this site is that K and KB equal 1000 bytes; M and MB equal 1000000 bytes; G and GB equal 1000000000 bytes, etc. The FORTRAN/COBOL notation 2**64 (rather than 2^64) is also preferred; see here for a discussion.


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PAPERS (Unpublished)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)






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.

Transcripts of these charts may be available from various other parties. I do not authorize, forbid, or restrict sales or distribution by such parties, known or unknown, for profit or not. Since I am not a participant or stakeholder in such operations, I do not accept legal responsibilty or liability for such sales or products.

Please note that I have declared all of my own Paydirt and Bowlbound charts, as well as all related developmental materials, to be in the public domain. However, certain hostile parties dispute my right to do this.

Team charts produced by other parties, whether or not based on my developmental materials, are their own intellectual, legal, and financial property, and are not subject to my declaration of public domain. They are entitled to their own copyrights and authorship notices.

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.

Please do not contact me regarding the Paydirt or Bowlbound charts or materials. The above exposition contains all that I have to say about the subject.


Martin Raab has discovered a new first (and largest) known occurrence prime gap of measure G=6582144 following the 216841-digit prime P1=499973#/30030 - 4509212 (where 499973# indicates the product of all primes from 2 through 499973 inclusive). This gap was first reported by Raab on 01 July 2017. The endpoints have passed the strong BPSW test (Nicely, 04 September 2017) for probabilistic primality. All the interior integers have been demonstrated composite (18 August 2017) by ATH on the Mersenne Forum. A test for deterministic certification of primality is at present out of the question. The gap has merit M=13.182884.


The Gapcoin network (Jonnie Frey, developer), a Bitcoin derivative which employs a hashing algorithm to search for prime gaps of high merit, has discovered a new prime gap of maximum known merit, a gap of G=8350 following the 87-digit prime P1=293703234068022590158723766104419463425709075574811762098588798217895728858676728143227. The merit M=G/ln(P1) of this gap is M=41.93878373153988, the largest merit of any known prime gap, and the first prime gap to be discovered with a merit exceeding 40. The endpoints of the gap have been certified as primes deterministically, using the Akiyama-Kida-O'Hara UBASIC implementation (1988-1992) of the APRCL2 test, due to Adleman, Pomerance, Rumely, Cohen, H. W. Lenstra, and A. K. Lenstra (1984-1987).

However, Bertil Nyman's maximal gap of 1132, following the prime 1693182318746371 (discovered 24 January 1999), continues to exhibit the greatest known value (0.92063858855742) of the Cramér-Shanks-Granville ratio G/ln²(p_1); this ratio is 0.210642105494715467 for the new Gapcoin gap. The limit superior of this ratio has been conjectured to be unity (or some even larger value); see the discussion in "New prime gaps between 1e15 and 5e16".

Thanks to Dana Jacobsen for alerting me to the discovery of this gap.

On 08 May 2019, Robert W. Smith discovered a new first known occurrence prime gap of 203890 following the 2485-digit prime 140207*5813#/46410 - 86644. This gap has merit 35.640174363, the greatest merit for any known prime gap exceeding 26892. On 08 July 2019, Smith also discovered a new first known occurrence prime gap of 614640 following the 10004-digit prime 281*23173#/46410 - 267338. This gap has merit 26.6845515588753865, the greatest merit for any known prime gap exceeding 556982. An extended table of previous such gaps, due to Robert W. Smith and axn, is available on the Mersenne PGS forum.


As a result of the continuing extension of the upper bound of exhaustive scans for prime gaps, the first known occurrence prime gaps of 1530 and 1550, following respectively the primes 17678654157568189057 and 18361375334787046697, and discovered respectively 19 April 2014 and 13 July 2014 by the late Dr. Bertil Nyman, have now been confirmed (13 August 2018) as first occurrence prime gaps and maximal prime gaps. Nyman's maximal prime gap of 1550 is the largest maximal prime gap presently known.

The merit of Nyman's new maximal prime gap G=1550 is M=34.9439.


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

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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.


Note that I have declared all of my work to be in the public domain.

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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. Furthermore, I find the current obsession with plagiarism and proprietary rights to be selfish, obstructive, and counterproductive.

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