Dr. Bertil Nyman

Uppsala, Sweden

http://www.trnicely.net

Current e-mail address

Last modified..............0700 GMT 05 April 2012. Journal citation...........Journal of Integer Sequences 6 (2003), Number 3, Article 03.3.1, pp. 1-6 (electronic). Mathematical Reviews.......MR1997838 (2004e:11143). Publication date...........13 August 2003. Accepted for publication...13 August 2003. Revision submitted.........13 August 2003. Original submission........10 February 2003.

Secondary: 11-04, 11Y55.

A *prime gap* G is the interval bounded by two consecutive prime
numbers q_k and q_(k+1). The *measure* (size, magnitude) g
of a prime gap G is the difference g=q_(k+1)-q_k of its bounding primes.
A prime gap is often specified by its measure g and its initial prime
p_1=q_k, and less often by the measure g and the terminal prime
p_2=q_(k+1). A prime gap of measure g contains g-1 consecutive
composite integers. The measures of the prime gaps are the successive
elements of the sequence D. Since two is the only even prime, every
prime gap is of even measure, with the sole exception of the prime gap of
measure 1 following the prime 2.

In illustration, a gap of measure g=6 (or simply a gap of 6) follows the prime p_1=23, while a gap of 10 follows the prime 139.

It is elementary that gaps of arbitrarily large measure exist, since, as observed by Lucas (1891), for n > 0 the integer (n+1)! + 1 must be followed by at least n consecutive composites, divisible successively by 2,3,...,n+1; however, n+1 represents only a lower bound on the measure of such gaps.

The *merit* M of a prime gap of measure g following the prime
p_1 is defined as M=g/ln(p_1). It is the ratio of the measure
of the gap to the "average" measure of gaps near that point; as a
consequence of the Prime Number Theorem, the average difference between
consecutive primes near x is approximately ln(x).

A prime gap of measure g is considered a *first occurrence prime
gap* when no smaller consecutive primes differ by exactly g, i.e., when
this is the first appearance of the positive integer g in the sequence
D. Thus, the gap of 4 following 7 is a first occurrence, while
the gap of 4 following 13 is not. Note that this usage of the
compound adjective *first occurrence* carries no implication whatsoever
regarding historical precedence of discovery. Multiple instances of gaps
of 1048 are known, but none is yet known to be a first occurrence, even
though one of them bears an earliest historical date of discovery. This
terminology follows that of Young and Potler (1989), and produces more
concise phrasing than some past and present alternative nomenclature.

A prime gap of measure g is titled *maximal* if it strictly exceeds
all preceding gaps, i.e., the difference between any two consecutive
smaller primes is < g, so that g exceeds all preceding elements of
D. Thus the gap of 6 following the prime 23 is a maximal prime gap,
since each and every smaller prime is followed by a gap less than 6
in measure; but the gap of 10 following the prime 139, while a first
occurrence, is not maximal, since a larger gap (the gap of 14 following
the prime 113) precedes it in the sequence of integers. Maximal prime
gaps are *ipso facto* first occurrence prime gaps as well.

Furthermore, the term *first known occurrence prime gap* is used to
denote a prime gap of measure g which has not yet been proven to be
(and may or may not be) the true first occurrence of a gap of measure
g; this situation arises from an incomplete knowledge of the gaps (and
primes) below the first known occurrence. Thus, Nyman discovered a
gap of 1048 following the prime 88089672331629091, and no smaller
instance is known; but since his exhaustive scan extended only to
5e16, this gap remains for the moment merely a first known occurrence,
not a first occurrence. First known occurrences serve as upper bounds
for first occurrences not yet established.

The search for first occurrence and maximal prime gaps was previously extended to 1e15 by the works of Glaisher (1877), Western (1934), Lehmer (1957), Gruenberger, Armerding, and Baker (1959, 1961), Appel and Rosser (1961), Lander and Parkin (1967), Brent (1973, 1980), Young and Potler (1989), and Nicely (1999). The present work extends this upper bound to 5e16. The calculations are currently being continued toward 4e18, by Tomás Oliveira e Silva (2001-2012), as part of a project generating numerical evidence for the Goldbach conjecture.

Among the measures taken to guard against errors (whether originating in logic, software, or hardware), the count pi(x) of primes was maintained and checked periodically against known values, such as those published by Riesel (1994), and especially the extensive values computed recently by Silva (2001-2012). In addition, Nicely has since duplicated Nyman's results through 4.5e15.

Listings of the 423 previously known first occurrence prime gaps (including 61 maximal gaps), those below 1e15, have been published collectively by Young and Potler (1989) and Nicely (1999), and are herein omitted for brevity.

A comprehensive listing of first occurrence and maximal prime gaps, annotated with additional information, is available at Nicely's URL. Nicely also maintains at his URL extensive lists of first known occurrence prime gaps, lying beyond the present upper bound of exhaustive computation, and discovered mostly by third parties, notably Harvey Dubner (1995-2003). These lists exhibit specific gaps for every even positive integer up to 10884, as well as for other scattered even integers up to 233822; for some of the gaps exceeding 8000 in magnitude, the bounding integers have only been proved strong probable primes (based on multiple Miller's tests).

The largest gap herein established as a first occurrence is the maximal gap of 1184 following the prime 43841547845541059, discovered 31 August 2002 by Nyman. The smallest gap whose first occurrence remains uncertain is the gap of 1048.

The maximal gap of 1132 following the prime 1693182318746371, discovered 24 January 1999 by Nyman, is the first occurrence of any "kilogap," i.e., any gap of measure 1000 or greater. Its maximality persists throughout an extraordinarily large interval; the succeeding maximal gap is the gap of 1184 following the prime 43841547845541059. The ratio of the initial primes of these two successive maximal gaps is 25.89, far exceeding the previous extreme ratio of 7.20 for the maximal gaps of 34 (following 1327) and 36 (following 9551), each discovered by Glaisher (1877). Furthermore, the gap of 1132 has the greatest merit (32.28) of any known gap; the maximal gap of 1184 is the only other one below 5e16 having a merit of 30 or greater.

The gap of 1132 is also of significance to the related conjectures put forth by Cramér (1936) and Shanks (1964), concerning the ratio g/ln²(p_1). Shanks reasoned that its limit, taken over all first occurrences, should be 1; Cramér argued that the limit superior, taken over all prime gaps, should be 1. Granville (1994), however, provides evidence that the limit superior is >= 2*exp(-gamma) = 1.1229. For the 1132 gap, the ratio is 0.9206, the largest value observed for any p_1 > 7, the previous best being 0.8311 for the maximal gap of 906 following the prime 218209405436543, discovered by Nicely (1999) in February, 1996. [See also the Addendum]

Several models have been proposed in an attempt to describe the distribution of first occurrence prime gaps, including efforts by Western (1934), Cramér (1936), Shanks (1964), Riesel (1994), Rodriguez (1999), Silva (2001-2012), and Wolf (1997); further details are available here. We simply note here Nicely's empirical observation that all first occurrence and maximal prime gaps below 5e16 obey the following relationship:

(1) 0.122985*sqrt(g)*exp(sqrt(g)) < p_1 < 2.096*g*exp(sqrt(g)) .

The validity of (1) for *all* first occurrence prime gaps
remains a matter of speculation. Among its corollaries would be the
conjecture that every positive even integer represents the difference
of some pair of consecutive primes, as well as a fairly precise estimate
for the answer to the question posed in 1964 by Paul A. Carlson to Daniel
Shanks (1964), to wit, the location of the first occurrence of one
million consecutive composite numbers. The argument g=1000002 entered
into (1) yields the result 2.4e436 < p_1 < 4.2e440, which is near
the middle of Shanks' own estimate of 1e300 < p_1 < 1e600.

- Kenneth I. Appel and J. Barkley Rosser, "Table for estimating functions of primes," Communications Research Division Technical Report Number 4, Institute for Defense Analyses, Princeton NJ (1961), xxxii + 125 pp. (22 cm.). Reviewed in Math. Comp. 16:80 (1962) 500-501 RMT 55.
- C. L. Baker and F. J. Gruenberger, "The first six million prime numbers," The Rand Corporation, Santa Monica CA, published by the Microcard Foundation, Madison WI (1959), 8 pp. (16x23 cm.) + 62 cards (7.5x12.6 cm). Reviewed in Math. Comp. 15:73 (1961) 82 RMT 4.
- Richard P. Brent, "The first occurrence of large gaps between successive primes," Math. Comp. 27:124 (1973), 959-963. MR 48 #8360.
- Richard P. Brent, "The first occurrence of certain large prime gaps," Math. Comp. 35:152 (1980), 1435-36. MR 81g:10002.
- Harald Cramér, "On the order of magnitude of the difference between consecutive prime numbers," Acta Arith. 2 (1936), 23-46.
- Marc Deléglise and Joël Rivat, "Computing pi(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method," Math. Comp. 65 (1996), 235-245. MR 96d:11139.
- Harvey Dubner, e-mail communications to Nicely (1995-2003).
- J. W. L. Glaisher, "On long successions of composite numbers," Messenger of Mathematics 7 (1877), 102-106, 171-176.
- Andrew Granville, "Unexpected irregularities in the distribution of prime numbers," in Proceedings of the International Congress of Mathematicians, Vol. I (Zürich, 1994), Birkhäuser, Basel, 1995, pp. 388-399. MR 97d:11139.
- F. J. Gruenberger and G. Armerding, "Statistics on the first six million prime numbers," Paper P-2460 of the Rand Corporation, Santa Monica CA (1961), 145 pp. (8.5" x 11"). Reviewed in Math. Comp. 19:91 (1965) 503-505 RMT 73.
- L. J. Lander and T. R. Parkin, "On the first appearance of prime differences," Math. Comp. 21 (1967), 483-488. MR 37#6237.
- Derrick Henry Lehmer, "Tables concerning the distribution of primes up to 37 millions," 1957. Copy deposited in the UMT file and reviewed in MTAC 13 (1959), 56-57.
- François Édouard Anatole Lucas, "Théorie des Nombres," Vol. 1, Gauthier-Villars, Paris, 1891, p. 360. Reprinted by A. Blanchard, Paris, 1961. MR 23#A828.
- Thomas R. Nicely, "New maximal prime gaps and first occurrences", Math. Comp. 68:227 (July, 1999), 1311-1315. MR1627813 (99i:11004). Electronic reprint available at http://www.trnicely.net/gaps/gaps.html.
- Paulo Ribenboim, "The New Book of Prime Number Records," 3rd ed., Springer-Verlag, New York, 1996, pp. 248-258. MR 96k:11112.
- Hans Riesel, "Prime Numbers and Computer Methods for Factorization," 2nd ed., Birkhäuser, Boston, 1994, pp. 78-82, 380-383. MR 95h:11142.
- Luis Rodriguez (AKA Luis Rodriguez Abreu/Torres), e-mail communication to Nicely (15/18 January 1999); also noted (August 2003) at http://www.utm.edu/research/primes/notes/errata/index.html.
- Daniel Shanks, "On maximal gaps between successive primes," Math. Comp. 18 (1964), 646-651. MR 29 #4745.
- Tomás Oliveira e Silva, electronic documents available (April,
2012) at
http://www.ieeta.pt/~tos/gaps.html. Computations carried out
2001-2012 in conjunction with Siegfried Herzog, Silvio Pardi,
*et al*. - A. E. Western, "Note on the magnitude of the difference between successive primes," J. London Math. Soc. 9 (1934), 276-278.
- Marek Wolf, "First occurrence of a given gap between consecutive primes", preprint (April, 1997), available (August, 2003) at http://www.ift.uni.wroc.pl/~mwolf.
- Jeff Young and Aaron Potler, "First occurrence prime gaps," Math. Comp. 52:185 (1989), 221-224. MR 89f:11019.
- Top of page
- "New maximal prime gaps and first occurrences"
- Listing of all known first occurrence and maximal prime gaps
- Tables of first known occurrence prime gaps
- Home page

============================================================================ TABLE 1. First occurrence prime gaps between 1e15 and 5e16 ============================================================================ Gap Following Gap Following Gap Following the prime the prime the prime ============================================================================ 796 1271309838631957 928 10244316228469423 1010 21743496643443551 812 1710270958551941 930 3877048405466683 1012 22972837749135871 824 1330854031506047 932 10676480515967939 1014 13206732046682519 838 1384201395984013 934 8775815387922523 1016 25488154987300883 842 1142191569235289 936 2053649128145117 1018 37967240836435909 846 1045130023589621 938 3945256745730569 1020 24873160697653789 848 2537070652896083 940 9438544090485889 1022 10501301105720969 850 2441387599467679 942 10369943471405191 1024 22790428875364879 852 1432204101894959 944 4698198022874969 1026 14337646064564951 854 1361832741886937 946 8445899254653313 1028 16608210365179331 856 1392892713537313 948 5806170698601659 1030 21028354658071549 858 1464551007952943 950 5000793739812263 1032 19449190302424919 864 2298355839009413 952 3441724070563411 1034 11453766801670289 866 2759317684446707 954 8909512917643439 1036 36077433695182153 868 1420178764273021 956 7664508840731297 1038 28269785077311409 870 1598729274799313 958 6074186033971933 1040 46246848392875127 874 1466977528790023 960 5146835719824811 1042 33215047653774409 876 1125406185245561 962 9492966874626647 1044 7123663452896833 878 2705074880971613 964 5241451254010087 1046 25702173876611591 882 3371055452381147 966 5158509484643071 1050 13893290219203981 884 1385684246418833 968 19124990244992669 1054 26014156620917407 886 4127074165753081 970 10048813989052669 1056 11765987635602143 888 2389167248757889 972 4452510040366189 1058 28642379760272723 890 3346735005760637 974 10773850897499933 1060 15114558265244791 892 2606748800671237 976 14954841632404033 1062 15500910867678727 894 2508853349189969 978 12040807275386881 1064 43614652195746623 896 3720181237979117 980 19403684901755939 1068 23900175352205171 898 4198168149492463 982 18730085806290949 1072 40433690575714297 900 2069461000669981 984 11666708491143997 1074 33288359939765017 902 1555616198548067 986 34847474118974633 1076 20931714475256591 904 3182353047511543 988 11678629605932719 1084 41762363147589283 908 2126985673135679 990 2764496039544377 1098 25016149672697549 910 1744027311944761 992 4941033906441539 1100 21475286713974413 912 2819939997576017 994 3614455901007619 1102 39793570504639117 914 3780822371661509 996 14693181579822451 1106 29835422457878441 *916 1189459969825483 998 11813551133888459 1108 43986327184963729 918 2406868929767921 1000 22439962446379651 1120 19182559946240569 920 4020057623095403 1002 14595374896200821 1122 31068473876462989 922 4286129201882221 1004 7548471163197917 *1132 1693182318746371 *924 1686994940955803 1006 37343192296558573 *1184 43841547845541059 926 6381944136489827 1008 5356763933625179 ============================================================================ *Maximal gap. ============================================================================

Nyman's gap of 1132 (merit 32.28254764) has been surpassed in merit by the maximal gap of 1476 following the prime 1425172824437699411 (merit 35.310308), discovered 01 April 2009 by Professor Tomás Oliveira e Silva, Universidade de Aveiro, Portugal; and also by Michiel Jansen, who discovered (03 January 2012) a first known occurrence prime gap of even greater merit---a gap of 66520, merit 35.4244594, following the prime 1931*1933#/7230 - 30244.

However, Nyman's 1132 gap continues to exhibit the greatest known value (0.9206386) of the Cramér-Shanks-Granville ratio g/ln²(p_1); this ratio is 0.8447275 for Silva's gap of 1476, 0.8483347 for the Herzog-Silva gap of 1442 following the prime 804212830686677669 (21 November 2005), and 0.8311258 for for Nicely's gap of 906. For Jansen's gap of 66520, R_csg is only 0.0188648876.

Note that if the Cramér-Shanks-Granville ratio is defined instead as g/ln²(p_2), the gaps following 2, 3, and 7 no longer require exclusion as exceptional cases, and the gaps of 1132, 1476, and 1442 exhibit the three greatest known values for this ratio, without exception (the values for these three gaps would remain unchanged to seven decimal digits).