Dr. Thomas R. Nicely

Tom Nicely began publishing lists of record prime gap data on his web site in the summer of 2000 and maintained the lists until his death in a car accident on 11th. September 2019. Prior to his retirement, Dr. Nicely was professor of mathematics at the University of Lynchburg. The Internet Archive’s “Wayback Machine” has captured the Univerity‘s Remembering Dr. Thomas Nicely page.

The text below has been copied from the web page **“First occurrence prime gaps”** served from Tom’s personal web site `www.trnicely.net`. The date of last publication was 20th August 2019 and the web site is no longer available but the page was captured on 18th. November 2019 by the “Wayback Machine” where the original page content remains available.

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The convention is followed here that the size (or measure) G of a prime gap is the difference of its two bounding primes; consequently a gap G contains (G - 1) consecutive composite integers. Some authorities (e.g., J. W. L. Glaisher, Daniel Shanks) have taken (G - 1) itself as the size of the gap, while others (e.g., Richard P. Brent) have specified the size of a gap by the parameter r = G/2. Furthermore, some authorities (e.g., Hans Riesel, Paul Leyland) have specified a gap by the terminating prime p_2, rather than the initiating prime p_1 as employed herein.

Note that the term first occurrence prime gap (of measure G) refers to that interval p <= x <= p+G for which (1) p and p+G are primes, (2) p+t is composite for each integer t=1,...,G-1, and (3) no smaller positive prime p possesses these properties. The entity might be more accurately described as the earliest or smallest occurrence of the gap G, but the terminology first occurrence prime gap is well established in the literature. It does not imply historical precedence; a gap of equal measure, bounded by larger primes, may have been previously known.

The term first known occurrence prime gap refers to a gap which satisifies conditions (1) and (2) above, but for which condition (3) has not been verified; there may exist a gap (at this time undiscovered, or at least unknown to me) of the same size at some unknown location between the current upper bound of exhaustive computation and the listed gap. For prime gap measures whose first occurrence is not yet known, first known occurrences serve as upper bounds; the current upper limit of exhaustive scans serves as the lower bound. Additional lists of first known occurrence prime gaps are maintained on this site; this is the only list which contains first occurrence or maximal prime gaps. Furthermore, I have written, and made available for download, a powerful UBASIC computer code designed for the discovery of new first known occurrence prime gaps.

In these tables, the size or measure of the gap (difference of the bounding primes) is shown in the first column (positions 1-6).

The second column (positions 7-11) indicates the classification of the gap, as explained below.

Position 7 will ordinarily be blank. If an asterisk (*) should appear, it indicates that the gap is a maximal gap, strictly exceeding in measure all the prime gaps preceding it (those between consecutive prime numbers smaller in magnitude). In this case, it will in addition always be a definite first occurrence and certified prime gap.

Position 8 is blank.

The character in position 9 is the type of the gap; all of the gaps in this document are conventional (common, classic, standard, regular, ordinary, normal) prime gaps, indicated by the letter "C"; in other words, consecutive prime numbers differing by the measure of the gap, as defined by conditions (1) and (2) alone from the above definition of first occurrence prime gaps. This is the default; if the term prime gap is used without further qualification or elaboration, it refers to a conventional prime gap. Additional lists are anticipated, enumerating other types of prime gaps.

The character in position 10 indicates the first occurrence status of the prime gap. The character "F" signifies that the gap has definitely been established (by an exhaustive scan to or beyond that point) as a first occurrence prime gap. The character "N" signifies that the gap is definitely not a first occurrence (a prior occurrence is known, with smaller bounding primes). The character "?" signifies that the gap is a first known occurrence (no such gap with smaller bounding primes has yet been found), but that it is not presently known if it is the first occurrence (i.e., whether or not a gap of equal measure with smaller bounding primes exists).

The character in position 11 indicates whether the bounding integers of the gap are certified primes ("C") or probabilistic primes ("P"). The bounding integers of certified gaps (also titled confirmed, conclusive, deterministic, definite, definitive, or proven) have been conclusively proven prime, using trial prime divisors to the square root of the prime, or some test such as APRCL2 (Adleman-Pomerance-Rumely-Cohen-Lenstra-Lenstra) or ECPP. The gap is probabilistic (also titled "Monte Carlo") if the bounding integers have only been shown statistically prime (with a probability extremely close to one), using, for example, the Miller-Rabin test with multiple bases, or the strong Baillie-PSW primality test. For even larger integers (several hundred digits or more), probabilistic tests are orders of magnitude faster than deterministic tests, but all tests become computationally expensive for primes having thousands of digits. I attempt to personally certify the endpoint primes of smaller gaps (up to 200 digits), and to verify probabilistically (BPSW test) the endpoint integers of gaps up to about 10,000 digits. For gaps with more than about 10,000 digits, I employ Fermat's Test with base 2 to verify that the endpoint integers are either primes or base-2 pseudoprimes.

The interior integers of each gap have been certified composite, deterministically (by myself, the discoverer, or a third party), usually by means of trial divisors, a Fermat test, or a Miller-Rabin test. However, if the character in position 11 is a "?" (classification code "C??"), the bounding integers are probable primes (using at least Fermat's Test, base 2), but the interior integers of the gap have not been verified all composite to the satisfaction of Thomas R. Nicely; consequently, there remains a significant possibility that such a gap may in fact be smaller in measure than indicated, due to the as yet undetected presence of an interior prime.

In all cases, ultimate responsibility for the verification of a gap rests with the discoverer(s).

The listed discoverers (abbreviation in the third column) and dates (year in the fourth column) reflect my best knowledge or estimate. If the actual date of discovery is not known, the date of publication or the date of the preprint is shown; if this is not known, an estimate is given. Some or all of the gaps occurring below 1e5 were almost certainly known prior to the indicated discovery, and the attribution of some of the other gaps below 1e8 is in doubt; for example, Appel and Rosser published significant (but incomplete) results for the gaps below 1e8 in 1961, the same year that Gruenberger and Armerding published definitive results to 104395289 from data generated by Baker and Gruenberger in 1959. In general, attribution is not provided for certification of a gap already established probabilistically; and in any case, the discovery date corresponds to the probabilistic verification of the gap.

The fifth column (positions 26-32) states a so-called figure of merit for the gap. This indicates how much larger the gap is than the average gap (approximately ln(x), as a consequence of the Prime Number Theorem) between primes near that point; the greater the merit, the more unusual the gap. The merit is computed as G/ln(p_1); variations in use (and at one time employed in these tables) include G/ln(p_2) and G/ln((p_1 + p_2)/2), where p_1 and p_2 are the initiating and terminating primes of the gap. For all but the first few gaps, the differences among these formulas are trivial; indeed, if the results are rounded to two decimal places, I have found no discrepancies in the resulting values for any gap exceeding 112.

NOTE: As a general rule, gaps with merits less than 1.000000 will not be included in these tables. Such gaps would be smaller than the average gap in that region.

The sixth column in the tables (positions 33-38) indicates the number of decimal digits in the initiating prime.

The seventh column (positions 41 and beyond) shows the initiating prime (smaller bounding integer) of the gap. initiating primes longer than 200 characters are abbreviated; as the lists grow, additional restrictions of this nature (such as a minimum acceptable merit) may eventually become necessary. Abbreviated primes are shown in the form 123456789012345678901234567890..., with a few (usually 25 or more) of the most significant digits shown, followed by an ellipsis "...". In addition, the file allgaps.dat is available for download. This contains the complete (no truncation or abbreviation of primes) specification of each and every recorded first occurrence, maximal, and first known occurrence prime gaps. It is a text file (WinDOS format), with one line per gap in standard format. Note that this file is nearly 10 MB in size, and contains extremely long lines which will challenge most editors and file utilities (it is intended primarily for software input or output). Furthermore, I have also made available for download the zipfile merits.zip, which contains a text file specifying the measure G and the merit M=G/ln(p_1) for all known first occurrence and first known occurrence prime gaps. This much smaller file (less than 1 MB) should be of additional assistance in determining whether or not some newly discovered gap constitutes a new first known occurrence.

INSTRUCTIONS FOR SUBMITTING PRIME GAPS: If you wish to submit lists of gaps for inclusion in this document, please send them as attachments consisting of zipped plain text files. Avoid proprietary formats such as Word, WordPerfect, Excel, TeX, PDF, or PostScript. Avoid embedding data in formats such as "rich text", HTML, XHTML, or XML. Multimedia submissions (video, audio, images, etc.) and social media submissions (Twitter, YouTube, Facebook, instant messaging, etc.) will not be accepted. Note that word processors and ordinary e-mail will usually destroy any formatting, and may corrupt very long lines (127 characters or more). Click here for my current e-mail address.

Note, however, that if your zipfiles exceed several MB in size, it is not advisable to send them as attachments, as you will risk mailbox overflow or bounceback, depending on restrictions enforced by the e-mail providers. For such large files, furnish instead a pointer to a Website from which I can download the file.

If possible, use the exact format employed for the gaps in this document, with the exception that you must specify the initating prime in its entirety, either by use of a simple formula (such as 10^999 + 7, using BASIC syntax; the factorial [!] and primorial [#] operators are also available) or by the complete decimal expansion of the integer. If line continuation is employed, use a trailing backslash "\" as the line continuation character. Even though the initiating prime may be abbreviated in some of the posted listings, complete specifications of the primes are necessary for validation of the gaps.

As an aternative, you may simply submit your gaps in the form of one gap (two entries) per line, gap size first, initiating prime p_1 following, separated by two or more spaces:

ggggg ppppppppppppppppppppp ggggg ppppppppppppppppppppp ggggg ppppppppppppppppppppp ..... .....................with the size of the gap and initiating prime each as an unbroken sequence of decimal digits (no commas, decimal points, spaces, or other internal grouping symbols embedded within a sequence of digits); one gap per line; no other symbols on the line. The initiating prime may be specified, as above, by a recognizable expression (e.g., 10^999 + 348957 or 2843# - 4759).

Be aware that there may be a delay of several days before your gaps are posted, while I am combining and processing multiple submissions. Additional instructions are provided for the formatting and submission of gaps of one million or greater (or having initiating primes of 100000 or more characters), which would otherwise overflow the standard format descibed above.

I have made available for download the zipfile cglp4.zip, featuring a DOS/Wintel executable which will check gap listings for validity, using probabilistic primality tests of extremely high reliability. Included are sample input and output files, source code (GNU C with GMP), and support routines. I strongly recommend the use of this code to check your listings prior to submission.

Gaps of 1000000 or greater would overflow the standard format, which is described in the principal explanatory notes and employed in the other lists of prime gaps at this site. Consequently, a special format is employed for such gaps. A fictional example is shown below:

999999999 C?P WCFields 2004 113.7447 9999999 1234567890123456789Submissions of such gaps may be in the above format (explained in detail below), or in the simpler format

ggggggggg ppppppppppwhere ggggggggg is the gap measure and ppppppppp is the initiating prime, specified as a literal integer, or in formula form. In either format, line continuation (as specified below) is optional in gap submissions.

The precise format specifications for "megagaps" are similar to those provided in the principal explanatory notes, with exceptions as follows (see the principal notes for further explanation of terms and concepts).

The measure of the gap is shown in positions 1-9, right justified using leading blanks.

The classifications of the gaps are shown in positions 10-14. Position 10 is an asterisk for maximal gaps, otherwise a blank. Position 11 is always blank.Position 12 is (in this table) always a "C", indicating an ordinary or common prime gap. Position 13 is ordinarily a "?", indicating that the gap is a first known occurrence, but that it is not known whether or not it is a true first occurrence. This character would be an "F" if the gap had been proven a first occurrence, or an "N" if it had been proven not a first occurrence. Position 14 is a "P" if the bounding primes are probabilistic, or a "C" if the bounding primes have been certified deterministically.

If position 14 is a "?" (classification code "C??"), the bounding integers are probable primes (primes or strong base-2 pseudoprimes), but the interior integers of the gap have not been verified all composite to the satisfaction of Thomas R. Nicely; consequently, there remains a significant possibility that such a gap may in fact be smaller in measure than indicated, due to the as yet undetected presence of an interior prime.

Position 15 is blank.

Positions 16-23 carry an eight-character abbreviation indicating the discoverer(s) of the gap, as provided in the accompanying key. Position 24 is blank.

Positions 25-28 indicate the year of discovery. Position 29 is blank.

Positions 30-37 indicate the merit of the gap, to four decimal places. Position 38 is blank.

Positions 39-45 indicate the number of decimal digits in the initiating prime. Positions 46 and 47 are blank.

The value of the initiating prime begins in position 48. This value must be specified in full in submissions, but for primes exceeding 200 digits or characters, the value shown in the table is truncated (some of the initiating primes exceed 100,000 decimal digits). Abbreviated primes are shown in the form 123456789012345678901234567890..., with a few (usually 25 or more) of the most significant digits shown, followed by an ellipsis "...".

In gap submissions, the initiating prime must be specified in full. If a formula is available, it can presumably be expressed within the 200-character limit. If the prime is a literal integer, of course, it may contain many thousands of digits. In this event, it may all be written to a single record in the file, or it may be written using line continuation, with a trailing backslash "\" as the continuation character. The line continuation format used by the author employs 200 digits per line, occupying positions 48-247 inclusive, with the trailing backslash in position 248, and blanks in positions 1-47 of continuation lines (this matches the location of the prime's most significant 200 digits in the first line).

This format is also to be used for gaps whose measure is less than 1000000, but whose initiating primes contain more than 99999 digits.

The file allgaps.dat is available for download. This contains the complete (no truncation or abbreviation of primes) specification of each and every first known occurrence prime gap. It is a text file (WinDOS format), with one line per gap in standard format. Note that this file is nearly 10 MB in size, and contains extremely long lines which will challenge most editors and file utilities (it is intended primarily for software input or output).

I have also made available the zipfile merits.zip, which contains a text file specifying the measure G and the merit M=G/ln(p_1) for all known first occurrence and first known occurrence prime gaps. This smaller file (less than 1 MB) should be of additional assistance in determining whether or not some newly discovered gap constitutes a new first known occurrence.

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