New prime gaps between 10^{15} and 5 × 10^{16}

Bertil Nyman SaabTech Systems AB Uppsala Sweden

                   bertil.nyman@saabtech.se

Thomas R. Nicely1 1113 Dandridge Drive Lynchburg, VA 24501-2231 USA

                     trnicely@hotmail.com
                    http://www.trnicely.net

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.

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 2 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.

1Corresponding author.

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 π(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 3 lists the newly discovered first occurrence prime gaps resulting from the present study; maximal gaps are indicated by a double dagger (‡). 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 π_{2}(5×10^{16}) = 47177404870103, the maximum argument for which this function has been evaluated. Nyman also obtained π(5×10^{16}) = 1336094767763971 for the corresponding count of primes; this is the largest value of x for which π(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

Gap Following the prime Gap Following the prime Gap Following 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

Table 1. First occurrence prime gaps between 10^{15} and 5 × 10^{16}. ‡ denotes a maximal gap

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/ ln^{2}(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 ≥ 2e^{−γ} ≈ 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.

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:

0.122985 ·\sqrt{g} · exp \sqrt{g} < p_{1} < 2.096 · g · exp \sqrt{g} .  (1)

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.

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2000 Mathematics Subject Classification: Primary 11A41; Secondary 11-04, 11Y55. Keywords: Prime gaps, maximal gaps, first occurrences, prime numbers, kilogaps, maximal prime gaps.

Received February 10, 2003; revised version received August 13, 2003. Published in Journal of Integer Sequences, August 13, 2003.

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