%------------------------------------------------------------------------------ % Beginning of gaps3h.tex % % AMS-LaTeX2e source file for the manuscript % % ``New prime gaps between 10^15 and 5*10^16'', % % originally titled % % ``First Occurrence Prime Gaps in 10^15 < p < 5*10^16'', % % by Bertil Nyman and Thomas R. Nicely. % % Original submission 10 February 2003 to Journal of Integer Sequences. % % This revision submitted 13 August 2003 to Journal of Integer Sequences. % %------------------------------------------------------------------------------ \documentclass[12pt,reqno]{amsart} \usepackage[usenames]{color} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} % \tolerance=250 \renewcommand{\,}{\hskip.03em\relax} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\ul}{\underbar} \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \def\NNB{1.598508912\times10^{15}} \def\CFKO{10884} \def\GFKO{233822} \def\NV{4.5\times10^{15}} \def\TILDE{\raise0.5ex\hbox{$\sim$}} \def\tmark{\tiny\raise1.5ex\hbox{T\kern-0.1em{M}}\normalsize} \def\rmark{\raise1.3ex\hbox{\tiny\kern0.2em{R}% \kern-0.92em\scriptsize\mathhexbox20D}\normalsize} \def\cplusplus{{C}\raise0.4ex\hbox{\tiny{++}}\normalsize} \def\A{\approx} \def\S{\slash} \def\D{\ddag} \DeclareMathOperator{\Li}{Li} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm {\LARGE\bf New prime gaps between $\bf 10^{15}$ and $\bf 5\times10^{16}$} \vskip 1cm \large Bertil Nyman\\ SaabTech Systems AB \\ Uppsala \\ Sweden \\ \href{mailto:bertil.nyman@saabtech.se}{\tt bertil.nyman@saabtech.se}\\ \ \\ Thomas R. Nicely\footnote[1]{Corresponding author.}\\ 1113 Dandridge Drive \\ Lynchburg, VA 24501-2231 \\ USA\\ \href{mailto:trnicely@hotmail.com}{\tt trnicely@hotmail.com}\\ \href{http://www.trnicely.net}{\tt http://www.trnicely.net}\\ \end{center} \vskip .2 in \begin{center} {\bf Abstract} \end{center} The interval from $10^{15}$ to $5\times10^{16}$ 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 \href{http://www.trnicely.net}{\tt http://www.trnicely.net}. \smallskip \section{Introduction} We restrict our discussion to the positive integers. Let $Q$ denote the sequence of prime numbers, $Q = \{2,3,5,7,11,\ldots,q_k,q_{k+1},\ldots\}$, and $D$ the sequence of differences of consecutive prime numbers, $D = \{1,2,2,4,\ldots,q_{k+1}-q_k,\ldots\}$. A {\it prime gap} $G$ is the interval bounded by two consecutive prime numbers $q_k$ and $q_{k+1}$. The {\it 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 $2$. 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 \cite{Lu}, for $n > 0$ the integer $(n+1)! + 1$ must be followed by at least $n$ consecutive composites, divisible successively by $2, 3, \ldots, n+1$; however, $n+1$ represents only a lower bound on the measure of such gaps. The {\it 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 {\it 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 {\it 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 \cite{YP}, and produces more concise phrasing than some past and present alternative nomenclature. A prime gap of measure $g$ is titled {\it 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 {\it ipso facto} first occurrence prime gaps as well. Furthermore, the term {\it 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 $5\times10^{16}$, 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 \cite{Gl}, Western \cite{We}, Lehmer \cite{Le}, Appel and Rosser \cite{AR}, Lander and Parkin \cite{LP}, Brent \cite{Br73,Br80}, Young and Potler \cite{YP}, and Nicely \cite{Ni99}. The present work extends this upper bound to $5\times10^{16}$. The calculations are currently being continued beyond $5\times10^{16}$ by Tom\'as Oliveira e Silva \cite{Silva}, as part of a project generating numerical evidence for the Goldbach conjecture. \section{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 $\NNB$ through $5\times10^{16}$. 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 $\NNB$, 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 \cite{Ri94}, and especially the extensive values computed recently by Silva \cite{Silva}. In addition, Nicely has since duplicated Nyman's results through $\NV$. \section{Computational Results} Table \ref{Table1} lists the newly discovered first occurrence prime gaps resulting from the present study; maximal gaps are indicated by a double dagger (\D). Each table entry shows the measure $g$ of the gap and the initial prime $p_1$. The fifteen gaps between $10^{15}$ and $\NNB$ are due to Nicely; all the rest were discovered by Nyman. \begin{table}[ht] \label{Table1} \begin{center} \vskip 1pc \begin{small} \begin{tabular}{||r|r||r|r||r|r||} \hline Gap&Following the prime&Gap&Following the prime&Gap&Following the prime\\ \hline 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\\ \D 916& 1189459969825483& 998& 11813551133888459& 1108& 43986327184963729\\ 918& 2406868929767921& 1000& 22439962446379651& 1120& 19182559946240569\\ 920& 4020057623095403& 1002& 14595374896200821& 1122& 31068473876462989\\ 922& 4286129201882221& 1004& 7548471163197917&\D1132& 1693182318746371\\ \D 924& 1686994940955803& 1006& 37343192296558573&\D1184& 43841547845541059\\ 926& 6381944136489827& 1008& 5356763933625179& & \\ \hline \end{tabular} \end{small} \end{center} \vskip .1in {\bf Table 1. First occurrence prime gaps between $\bf 10^{15}$ and $\bf 5\times10^{16}$. {\rm $\D$} denotes a maximal gap} \end{table} \section{Observations} As a collateral result of his calculations, Nyman has computed for the count of twin primes the value $\pi_2(5\times10^{16})=47177404870103$, the maximum argument for which this function has been evaluated. Nyman also obtained $\pi(5\times10^{16})=1336094767763971$ 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 Del\'eglise and Rivat \cite{DR}, using indirect sieving methods. Nyman has also generated frequency tables for the distribution of all prime gaps below $5\times10^{16}$. 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 \cite{YP} and Nicely \cite{Ni99}, 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 \cite{Du}. These lists exhibit specific gaps for every even positive integer up to $\CFKO$, as well as for other scattered even integers up to $\GFKO$; 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 $\A 25.89$, far exceeding the previous extreme ratio of $\A 7.20$ for the maximal gaps of $34$ (following $1327$) and $36$ (following $9551$), each discovered by Glaisher \cite{Gl} in $1877$. Furthermore, the gap of $1132$ has the greatest merit ($\A 32.28$) of any known gap; the maximal gap of $1184$ is the only other one below $5\times10^{16}$ having a merit of $30$ or greater. The gap of $1132$ is also of significance to the related conjectures put forth by Cram\'er \cite{Cr} and Shanks \cite{Sh64}, concerning the ratio $g/\ln^2(p_1)$. Shanks reasoned that its limit, taken over all first occurrences, should be 1; Cram\'er argued that the limit superior, taken over all prime gaps, should be 1. Granville \cite{Gr}, however, provides evidence that the limit superior is $\ge 2e^{-\gamma} \A 1.1229$. For the $1132$ gap, the ratio is $\A 0.9206$, the largest value observed for any $p_1 > 7$, the previous best being $\A 0.8311$ for the maximal gap of $906$ following the prime $218209405436543$, discovered by Nicely \cite{Ni99} in February, $1996$. Several models have been proposed in an attempt to describe the distribution of first occurrence prime gaps, including efforts by Western \cite{We}, Cram\'er \cite{Cr}, Shanks \cite{Sh64}, Riesel \cite{Ri94}, Rodriguez \cite{LR}, Silva \cite{Silva}, and Wolf \cite{MW}. We simply note here Nicely's empirical observation that all first occurrence and maximal prime gaps below $5\times10^{16}$ obey the following relationship: \vskip -10pt \begin{equation}\label{G1} 0.122985\cdot\mskip-2mu\sqrt{g}\cdot\exp{\sqrt{g}} \mskip9mu < \mskip8mu p_1 \mskip4mu < \mskip8mu 2.096\cdot g\cdot\exp{\sqrt{g}} \quad . \end{equation} \vskip 2pt The validity of \eqref{G1} for {\it 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 \cite{Sh64}, to wit, the location of the first occurrence of one million consecutive composite numbers. The argument $g=1000002$ entered into \eqref{G1} yields the result $2.4 \times 10^{436} < \mskip4mu p_1 < \mskip2mu 4.2 \times 10^{440}$, which is near the middle of Shanks' own estimate of $10^{300} < \mskip4mu p_1 < \mskip2mu 10^{600}$. \section{Acknowledgments} Nyman wishes to thank \hbox{SaabTech} Systems~AB for providing excellent computing facilities. \bibliographystyle{amsplain} \begin{thebibliography}{29} \bibitem {AR} Kenneth I. Appel and J. Barkley Rosser, Table for estimating functions of primes, IDA-CRD Technical Report Number 4 (1961). Reviewed in RMT \textbf{55}, \textit{Math. Comp.} \textbf{16} (1962), 500--501. \bibitem {Br73} Richard P. Brent, The first occurrence of large gaps between successive primes, \textit{Math. Comp.} \textbf{27} (1973), 959--963. MR \textbf{48\#}8360. \bibitem {Br80} Richard P. Brent, The first occurrence of certain large prime gaps, \textit{Math. Comp.} \textbf{35} (1980), 1435--36. MR \textbf{81g:}10002. \bibitem {Cr} Harald Cram\'er, On the order of magnitude of the difference between consecutive prime numbers, \textit{Acta Arith.} \textbf{2} (1936), 23--46. \bibitem {DR} Marc Del\'eglise and Jo\"el Rivat, Computing $\pi(x)$: the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, \textit{Math. Comp.} \textbf{65} (1996), 235--245. MR \textbf{96d:}11139. \bibitem {Du} Harvey Dubner, e-mail communications to Nicely (1995--2003). \bibitem {Gl} J. W. L. Glaisher, On long successions of composite numbers, \textit{Messenger of Mathematics} \textbf{7} (1877), 102--106, 171--176. \bibitem {Gr} Andrew Granville, Unexpected irregularities in the distribution of prime numbers, in \textit{Proceedings of the International Congress of Mathematicians, Vol. I (Z\"urich, 1994)}, Birkh\"auser, Basel, 1995, pp. 388--399. MR \textbf{97d:}11139. \bibitem {LP} L. J. Lander and Thomas R. Parkin, On first appearance of prime differences, \textit{Math. Comp.} \textbf{21} (1967), 483--488. MR \textbf{37\#}6237. \bibitem {Le} Derrick Henry Lehmer, Tables concerning the distribution of primes up to 37 millions (1957). Copy deposited in the UMT file and reviewed in \textit{MTAC} \textbf{13} (1959), 56--57. \bibitem {Lu} Fran\c{c}ois \'Edouard Anatole Lucas, \textit{Th\'eorie des Nombres}, Vol. 1, Gauthier-Villars, Paris, 1891, p. 360. Reprinted by A. Blanchard, Paris, 1961. MR \textbf{23\#}A828. \bibitem {Ni99} Thomas R. Nicely, New maximal prime gaps and first occurrences, \textit{Math. Comp.} \textbf{68} (July, 1999), 1311--1315. MR \textbf{99i:}11004. \bibitem {Rib} Paulo Ribenboim, \textit{The New Book of Prime Number Records}, 3rd ed., Springer-Verlag, New York, 1996, pp. 248--258. MR \textbf{96k:}11112. \bibitem {Ri94} Hans Riesel, \textit{Prime Numbers and Computer Methods for Factorization}, 2nd ed., Birkh\"auser, Boston, 1994, pp. 78--82, 380--383. MR \textbf{95h:}11142. \bibitem {LR} Luis Rodriguez (AKA Luis Rodriguez Abreu/Torres), e-mail communications to Nicely (15--18 January 1999). \bibitem {Sh64} Daniel Shanks, On maximal gaps between successive primes, \textit{Math. Comp.} \textbf{18} (1964), 646--651. MR \textbf{29\#}4745. \bibitem{Silva} Tom\'as Oliveira e Silva, electronic documents available (August, 2003) at \href{http://www.ieeta.pt/~tos/hobbies.html}{\tt http:/{\S}www.ieeta.pt{\S}{\TILDE}tos{\S}hobbies.html}. \bibitem {We} A. E. Western, Note on the magnitude of the difference between successive primes, \textit{J. London Math. Soc.} \textbf{9} (1934), 276--278. \bibitem {MW} Marek Wolf, First occurrence of a given gap between consecutive primes, preprint (April, 1997). Available (August, 2003) at \href{http://www.ift.uni.wroc.pl/~mwolf}{\tt http:/{\S}www.ift.uni.wroc.pl{\S}{\TILDE}mwolf}. \bibitem {YP} Jeff Young and Aaron Potler, First occurrence prime gaps, \textit{Math. Comp.} \textbf{52} (1989), 221--224. MR \textbf{89f:}11019. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11A41; Secondary 11-04, 11Y55. \noindent \emph{Keywords:} Prime gaps, maximal gaps, first occurrences, prime numbers, kilogaps, maximal prime gaps. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received February 10, 2003; revised version received August 13, 2003. Published in {\it Journal of Integer Sequences}, August 13, 2003. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}. \vskip .1in \end{document}