First known occurrence prime gaps
(1000000 to 999999998)
Thomas R. Nicely
Freeware copyright © 2013 Thomas R. Nicely. Released into the public
domain by the author, who disclaims any legal liability arising from
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.
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
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
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
The measure of the gap is shown in positions 1-9, right justified using
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
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
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