First known occurrence prime gaps (1000000 to 999999998)

First known occurrence prime gaps
(1000000 to 999999998)
Thomas R. Nicely

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

Last updated 0503 GMT 09 March 2013. Additional lists of prime gaps are maintained on this site.

For detailed explanatory notes, a complete bibliography, and a list of the discoverers and their associated abbreviations, see the document First occurrence prime gaps.

FORMAT

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  1234567890123456789

Submissions of such gaps may be in the above format (explained in detail below), or in the simpler format

ggggggggg  pppppppppp

where 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 strong probable primes, 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 (due to an available connection speed of only 56 Kbits/sec max). 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 "...".
A file containing the complete specifications of abbreviated primes, and all recorded first occurrence, maximal, and first known occurrence prime gaps, is nearly 20 MB in size. My dialup connection renders it impractical to maintain this file online and updated. However, I have 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 file should be of assistance in determining whether or not some newly discovered gap constitutes a new first known occurrence.
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.

============================================================================
    Gap    Cls Discvrer Year   Merit   Digits  Following the prime
============================================================================
  1001548  C?P RosntlJA 2004  10.0157   43429  1913094464943476849014660...
  1078180  C?P PierCami 2006  12.3203   38007  50491*(87811#)/6 - 657714
  1113106  C?P MJPC&JKA 2013  25.9045   18662  587*43103#/2310 - 455704
  1569660  C?P M.Jansen 2012   7.0001   97384  224737#/510510 - 1054198
  1575828  C?P M.Jansen 2012  15.0964   45334  104729#/2310 - 1282742
  2055816  C?P PierCami 2010  15.6743   56962  6887*(131591#)/2730 - 1381994
  2254930  C?P RosntlJA 2004  11.2755   86853  1122483511942968776411893...
  2724214  C?P MJandJKA 2013  11.8311  100000  230563#/2310 - 44352
  2765878  C?P MJPC&JKA 2013  12.0114  100006  230567#/2310 - 939244
  3311852  C?P MJandJKA 2012  14.6838   97953  226007#/2310 - 2305218
============================================================================