New prime gaps between 10^15 and 5*10^16

Dr. Bertil Nyman

SaabTech Systems AB
Uppsala, Sweden

Thomas R. Nicely

(Corresponding author)
http://www.trnicely.net
Current e-mail address

Copyright © 2007 Bertil Nyman and Thomas R. Nicely. All rights reserved. This document may be freely reproduced and distributed for educational and non-profit purposes.

Document History 

Last modified....................0530 EDT 28 March 2007.
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.

The content of this document is essentially that of the journal article as published; it may contain minor revisions and corrections, and differ in format and detail. The journal article is available in various formats (PS, PDF, dvi, AMS-LaTeX2e) at the home page of the Journal of Integer Sequences.

Abstract

The interval from 10^15 to 5*10^16 was searched for first occurrence prime gaps and maximal prime gaps. One hundred and twenty-two new first occurrences were found, including four new maximal gaps, leaving 1048 as the smallest gap whose first occurrence remains uncertain. The first occurrence of any prime gap of 1000 or greater was found to be the maximal gap of 1132 following the prime 1693182318746371. A maximal gap of 1184 follows the prime 43841547845541059. More extensive tables of prime gaps are maintained at http://www.trnicely.net.

Mathematics Subject Classification 2000 (MSC2000)

Primary: 11A41.
Secondary: 11-04, 11Y55.

Key Words and Phrases

Prime gaps, maximal gaps, first occurrences, prime numbers, kilogaps, maximal prime gaps.

1. Introduction

We restrict our discussion to the positive integers. Let Q denote the sequence of prime numbers, Q = {2,3,5,7,11,...,q_k,q_(k+1),...}, and D the sequence of differences of consecutive prime numbers, D = {1,2,2,4,...,q_(k+1)-q_k,...}.

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 [11], 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 [20], 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 5*10^16, 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 10^15 by the works of Glaisher [7], Western [18], Lehmer [10], Appel and Rosser [1], Lander and Parkin [9], Brent [2, 3], Young and Potler [20], and Nicely [12]. The present work extends this upper bound to 5*10^16. The calculations are currently being continued beyond 5*10^16, by Tomás Oliveira e Silva [17], as part of a project generating numerical evidence for the Goldbach conjecture.

2. Computational Technique

The calculations were carried out over a period of years, distributed asynchronously among numerous personal computers, taking advantage of otherwise idle CPU time. Nyman accomplished the bulk of the computations; employing as many as eighty systems from 1998 to 2002, he accounted for the survey of the region from 1.598508912*10^15 through 5*10^16. Nicely's enumerations of prime gaps began in the summer of 1995, but the portion reported here was carried out from 1997 to 1999, over the interval from 10^15 to 1.598508912*10^15, the number of systems in use varying from about five to twenty-five. The algorithms employed the classic sieve of Eratosthenes, with the addition of a few speed enhancing optimizations, to carry out an exhaustive generation and analysis of the differences between consecutive primes. More sophisticated techniques for locating large prime gaps, such as scanning through arithmetic progressions, were rendered impractical by the fact that the search for first occurrences was being carried out concurrently with other tasks; Nicely was enumerating prime constellations, while Nyman was gathering comprehensive statistics on the frequency distribution of prime gaps.

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 [14], and especially the extensive values computed recently by Silva [17]. In addition, Nicely has since duplicated Nyman's results through 4.5*10^15.

3. Computational Results

Table 1 lists the newly discovered first occurrence prime gaps resulting from the present study; maximal gaps are indicated by an asterisk (*). Each table entry shows the measure g of the gap and the initial prime p_1. The fifteen gaps between 10^15 and 1.598508912*10^15 are due to Nicely; all the rest were discovered by Nyman.

4. Observations

As a collateral result of his calculations, Nyman has computed for the count of twin primes the value pi_2(5*10^16) = 47177404870103, the maximum argument for which this function has been evaluated. Nyman also obtained pi(5*10^16) = 1336094767763971 for the corresponding count of primes; this is the largest value of x for which pi(x) has been determined by direct enumeration, and confirms the value previously obtained by Deléglise and Rivat [5], using indirect sieving methods. Nyman has also generated frequency tables for the distribution of all prime gaps below 5*10^16.

Listings of the 423 previously known first occurrence prime gaps (including 61 maximal gaps), those below 10^15, have been published collectively by Young and Potler [20] and Nicely [12], 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 [6]. 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 [7] in 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 5*10^16 having a merit of 30 or greater.

The gap of 1132 is also of significance to the related conjectures put forth by Cramér [4] and Shanks [16], 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 [8], 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 [12] 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 [18], Cramér [4], Shanks [16], Riesel [14], Rodriguez [15], Silva [17], and Wolf [19]. We simply note here Nicely's empirical observation that all first occurrence and maximal prime gaps below 5*10^16 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 [16], to wit, the location of the first occurrence of one million consecutive composite numbers. The argument g=1000002 entered into (1) yields the result 2.4*10^436 < p_1 < 4.2*10^440, which is near the middle of Shanks' own estimate of 10^300 < p_1 < 10^600.

5. Acknowledgments

Nyman wishes to thank SaabTech Systems AB for providing excellent computing facilities.

References

  1. Kenneth I. Appel and J. Barkley Rosser, "Table for estimating functions of primes," IDA-CRD Technical Report Number 4, 1961. Reviewed in RMT 55, Math. Comp. 16 (1962), 500-501.
  2. Richard P. Brent, "The first occurrence of large gaps between successive primes," Math. Comp. 27:124 (1973), 959-963. MR 48 #8360.
  3. Richard P. Brent, "The first occurrence of certain large prime gaps," Math. Comp. 35:152 (1980), 1435-36. MR 81g:10002.
  4. Harald Cramér, "On the order of magnitude of the difference between consecutive prime numbers," Acta Arith. 2 (1936), 23-46.
  5. Marc Deléglise and Joël Rivat, "Computing pi(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method," Math. Comp. 65 (1996), 235-245. MR 96d:11139.
  6. Harvey Dubner, e-mail communications to Nicely (1995-2003).
  7. J. W. L. Glaisher, "On long successions of composite numbers," Messenger of Mathematics 7 (1877), 102-106, 171-176.
  8. 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.
  9. L. J. Lander and T. R. Parkin, "On the first appearance of prime differences," Math. Comp. 21 (1967), 483-488. MR 37#6237.
  10. 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.
  11. François Édouard Anatole Lucas, "Théorie des Nombres," Vol. 1, Gauthier-Villars, Paris, 1891, p. 360. Reprinted by A. Blanchard, Paris, 1961. MR 23#A828.
  12. 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.
  13. Paulo Ribenboim, "The New Book of Prime Number Records," 3rd ed., Springer-Verlag, New York, 1996, pp. 248-258. MR 96k:11112.
  14. Hans Riesel, "Prime Numbers and Computer Methods for Factorization," 2nd ed., Birkhäuser, Boston, 1994, pp. 78-82, 380-383. MR 95h:11142.
  15. 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.
  16. Daniel Shanks, "On maximal gaps between successive primes," Math. Comp. 18 (1964), 646-651. MR 29 #4745.
  17. Tomás Oliveira e Silva, electronic documents available (August, 2003) at http://www.ieeta.pt/~tos/hobbies.html.
  18. A. E. Western, "Note on the magnitude of the difference between successive primes," J. London Math. Soc. 9 (1934), 276-278.
  19. 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.
  20. Jeff Young and Aaron Potler, "First occurrence prime gaps," Math. Comp. 52:185 (1989), 221-224. MR 89f:11019.

       TABLE 1. First occurrence prime gaps between 10^15 and 5*10^16

 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.

Addendum

NOTE: This addendum was not part of the submission or the publication.

Nyman's gap of 1132 (merit 32.28254764) has been surpassed in merit by the gap of 1442 following the prime 804212830686677669 (merit 34.9756865), discovered circa 21 November 2005 by Professor Siegfried "Zig" Herzog of Penn State University (Mont Alto), using computer codes developed by Professor Tomás Oliveira e Silva, Universidade de Aveiro, Portugal.

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 0.8483347 for the Herzog-Silva 1442 gap, and 0.8311258 for Nicely's 906 gap.

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, 1442, and 906 exhibit the three greatest known values for this ratio, without exception (the values for these three gaps would remain unchanged to seven decimal places).