Some Results of Research in Computational Number Theory

Dr. Thomas R. Nicely

Current e-mail address

Site last updated 0600 GMT 12 June 2017.

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.


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PAPERS (Unpublished)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 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.


Robert W. Smith (Oxford, UK) has discovered a new first (and largest) known occurrence prime gap of measure G=5103138 following the 216849-digit prime P1=281*499979#/46410 - 2702372 (where 499979# indicates the product of all primes from 2 through 499979 inclusive). This gap was first reported by Smith on 08 August 2016, with double checking concluded by Martin Raab and "ATH" (22 October 2016). The endpoints have passed the BPSW test for probabilistic primality (no BPSW pseudoprimes are known, and it is known that none exist below 2^64). A test for deterministic certification of primality is at present impractical. The gap has merit M=10.2203.


Dana Jacobsen has discovered (08 January 2017) a new first known occurrence prime gap of measure G=18306 following the 209-digit prime P1=650094367*491#/2310 - 8936 (where 491# indicates the product of all primes from 2 through 491 inclusive). The merit M=G/ln(P1) of this gap is M=38.06646, the largest merit of any known prime gap. 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 (0.9206386) of the Cramér-Shanks-Granville ratio G/ln²(p_1); this ratio is 0.07915948 for Jacobsen'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 5e16".


Professor Tomás Oliveira e Silva (Universidade de Aveiro, Portugal) and Professor Siegfried "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). The merit of the maximal gap G=1476 is M=35.3103.


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, 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.



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.


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