The Top-20 Prime Gaps

Record tables below:
Top-20 merits
Top-20 gaps with merit above 10
Top-20 gaps with merit above 20
Gaps which are the largest with at least that merit
On subpages:
Gaps which are among the 20 largest with at least that merit
Maximal prime gaps

Definitions for this site:
There is a prime gap with positive integers p1 and p2 as end points, if p1 < p2 are consecutive primes (all intermediate numbers are composites). Some people define p1+1 and p2-1 to be the end points.
The size of the prime gap is p2 - p1. Some people define it to be one less.
The merit of the prime gap is size / ln p1, where ln is the natural logarithm. Some people use  p2 or a number between p1 and p2, but the difference is microscopic for large primes.

This site requires that all numbers inside a listed prime gap have been proved to be composite, but the end points are not required to be proven primes. If they are not proven then they must be probable primes, also called PRP's. Specifically they must have passed at least 5 Miller-Rabin tests or Fermat PRP tests with different bases, or stronger PRP tests.
A PRP can loosely speaking be considered "almost certainly" prime, based on statistical properties of PRP tests, but there is a small risk that a PRP is actually composite (very small for large PRP's or many PRP tests). In that case, the gap listed here would just be part of an even larger prime gap which would still qualify for the tables, possibly at a better position. PRP's are accepted here and on some other prime gap pages for this reason, although PRP's are often not accepted in other contexts, e.g. lists of the largest known primes.
Proven primes are preferred here when practical, but prime gap searches usually produce PRP's which are not easily provable when the PRP is large. If the whole top-20 table with merit above 10 or 20 is PRP's then the single largest gap with proven end points is added without rank.

The average prime gap near an integer N is approximately ln N. The merit indicates the relative size of a prime gap, compared to the approximate average for that size primes. This site only accepts prime gaps with merit above 10.0 (and so, in a loose sense, the gap must be at least 10 times larger than is typical). At the opposite end, the smallest known merit as of 2011 is achieved for the largest known twin prime, 3756801695685*2^666669+/-1 with 200700 digits and merit 0.000004328, found by Timothy D. Winslow, PrimeGrid, TwinGen, LLR.

Thomas R. Nicely has composed tables of First occurrence prime gaps and first known occurrence prime gaps. These tables include most or all prime gaps in the tables below, and many more. He uses the same definitions and his tables may sometimes be more up to date than mine.

Please mail me with new candidates for the tables, or corrections if you think there are errors. Indicate whether the end points are proven primes and give a simple expression if possible. When multiple gaps are submitted, Nicely's format is preferred. I will strive to update within 2 days of receiving a submission. If a gap is only submitted to Nicely then it should eventually turn up here but it may take a while.

The person running a program is credited as discoverer. If a specialized prime gap program is used then the programmer is listed afterwards, when known. A general program (not designed for large prime gaps) such as a sieve, PRP tester or primality prover is usually not mentioned. The original top-20 page and Nicely's site do not mention these programs and this site follows what might be called the prime gap practice.
For the record: All gaps involving me (Jens K. Andersen) used my own sieve and either the GMP library (usually below 1100 digits) or PrimeForm/GW for PRP testing. Marcel Martin's Primo proved all proven end points, except the gap of 337446 with 7996-digit primes which were proved by François Morain with fastECPP.

In 1931 E. Westzynthius proved there are arbitrarily large merits, i.e. for any m there exist gaps with merit > m.
A rough heuristical estimate which may deteriorate for large m indicates around 1 in em prime gaps has merit > m. e10 ~= 20000, e20 ~= 5·108, e30 ~= 1013. It is possible to greatly increase these odds in gap searches among carefully selected large numbers, by using modular equations to ensure unusually many numbers with a small factor. Unfortunately the best methods produce numbers with no simple expression.
There are usually only few prime gaps with simple expressions for the end points among the 20 largest gaps for any merit. However, the single largest gap with "basic" expression and merit above 10 or 20 is listed in those tables, without rank if outside the top-20. A basic expression is here defined as maximum 25 characters, all taken from 0123456789+-*/^( ). Primorial and factorial are not allowed since they can be used to ensure many small factors, and the idea of the basic expression record is partly to avoid special prime gap methods.

Big decimal expansions are in a separate file, or will be available by e-mail request. P838 means 838-digit end points which are proven primes. PRP43429 means one or two PRP end points with 43429 digits. The digit count is for the gap start p1 if there is a difference. n# (called n primorial) is the product of all primes ≤ n, e.g. 7# = 2 · 3 · 5 · 7.

 

Top-20 merits
Rank Size Gap start Merit Discoverer Year
1 10716 P127 = 7910896513*283#/30 - 6480 36.86     Dana Jacobsen 2016
2 13692 P163 = 1037600971*383#/210 - 8776 36.59     Dana Jacobsen 2016
3 26892 P321 = 59740589*757#/210 - 14302 36.42     Dana Jacobsen 2016
4 66520 P816 = 1931*1933#/7230 - 30244 35.42     Michiel Jansen 2012
5 1476 P19 = 1425172824437699411 35.3103 Tomás Oliveira e Silva 2009
6 1442 P18 = 804212830686677669 34.9757 Siegfried Herzog & Tomás Oliveira e Silva 2005
7 1550 P20 = 18361375334787046697 34.9439 Bertil Nyman 2014
8 18840 P236 = 962682899*563#/30 - 3918 34.76     Dana Jacobsen 2016
9 11350 P142 = 693236519*337#/210 - 2778 34.75     Dana Jacobsen 2016
10 12176 P153 = 577924573*359#/30 - 7508 34.61     Dana Jacobsen 2016
11 1530 P20 = 17678654157568189057 34.5225 Bertil Nyman 2014
12 14614 P185 = 898651*443#/7710 - 7186 34.48     Dana Jacobsen 2015
13 12510 P159 = 1584331571*373#/30 - 8020 34.30     Dana Jacobsen 2016
14 11868 P151 = 1248652829*353#/30 - 8222 34.23     Dana Jacobsen 2016
15 22704 P289 = 18961493*677#/210 - 12370 34.15     Dana Jacobsen 2016
16 1454 P19 = 3219107182492871783 34.1189 Silvio Pardi & Tomás Oliveira e Silva 2011
17 11160 P143 = 1210091941*337#/210 - 6598 34.11     Dana Jacobsen 2016
18 13782 P177 = 1602369251*419#/30 - 4372 33.90     Dana Jacobsen 2016
19 4970 P64 = 5003008567... 33.8849 Helmut Spielauer 2015
20 2258 P29 = 89131646821029739238184261281 33.8734 Helmut Spielauer 2016

 

Top-20 gaps with merit above 10
Rank Size Gap start Merit Discoverer Year
1 5103138 PRP216849 = 281*499979#/46410 - 2702372 10.22     Robert Smith 2016
2 4680156 PRP99750 = 230077#/2229464046810 - 3131794 20.38     Martin Raab 2016
3 3311852 PRP97953 = 226007#/2310 - 2305218 14.68     Michiel Jansen & Jens K. Andersen 2012
4 2945060 PRP99750 = 230077#/2227174919970 - 1072622 12.82     Martin Raab 2016
5 2765878 PRP100006 = 230567#/2310 - 939244 12.01     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
6 2724214 PRP100000 = 230563#/2310 - 44352 11.83     Michiel Jansen & Jens K. Andersen 2013
7 2586246 PRP99750 = 230077#/2226554139810 - 1616044 11.26     Martin Raab 2016
8 2559528 PRP99750 = 230077#/2228280684630 - 961364 11.14     Martin Raab 2016
9 2493532 PRP99750 = 230077#/2227349514390 - 1441218 10.86     Martin Raab 2016
10 2435476 PRP99750 = 230077#/2226961526790 - 994222 10.60     Martin Raab 2016
11 2254930 PRP86853 = 11224835119429687764118930339136... 11.28     Hans Rosenthal & Jens K. Andersen 2004
12 2055816 PRP56962 = 6887*(131591#)/2730 - 1381994 15.67     Pierre Cami 2010
13 1575828 PRP45334 = 104729#/2310 - 1282742 15.10     Michiel Jansen 2012
14 1462522 PRP39448 = 91229#/46056680670 - 853776 16.10     Martin Raab 2015
15 1452592 PRP43291 = 451*99991#/46410 - 1136946 14.57     Robert Smith 2016
16 1286500 PRP21608 = 1111111111111111111*49999#/(510510*499) - 525318 25.86     Hans Rosenthal 2016
17 1224222 PRP43293 = 8263*99991#/46410 - 600164 12.28     Robert Smith 2016
18 1217460 PRP39448 = 91229#/46093437390 - 495038 13.40     Martin Raab 2015
19 1176666 PRP39443 = 91199#/46473256830 - 547454 12.96     Martin Raab 2015
20 1117820 PRP43291 = 149*99991#/46410 - 813426 11.21     Robert Smith 2016
Largest gap with proven end points:
-- 1113106 P18662 = 587*43103#/2310 - 455704 25.90     Michiel Jansen, Pierre Cami, Jens K. Andersen, Primo 2013
Largest gap with basic expression:
-- 725724 PRP31103 = 10^31103 - 86991 10.13     Patrick De Geest 2008

 

Top-20 gaps with merit above 20
Rank Size Gap start Merit Discoverer Year
1 4680156 PRP99750 = 230077#/2229464046810 - 3131794 20.38     Martin Raab 2016
2 1286500 PRP21608 = 1111111111111111111*49999#/(510510*499) - 525318 25.86     Hans Rosenthal 2016
3 1113106 P18662 = 587*43103#/2310 - 455704 25.90     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
4 984108 PRP16901 = 39161#/2310 - 510478 25.29     Michiel Jansen 2012
5 973764 PRP18648 = 431*43063#/2310 - 278398 22.68     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
6 865056 PRP18636 = 44633*(43037#)/2310 - 394442 20.16     Pierre Cami 2013
7 834114 PRP16497 = 38231#/2310 - 393706 21.96     Michiel Jansen 2012
8 818886 PRP17071 = 39541#/2310 - 380216 20.83     Michiel Jansen 2012
9 784246 PRP16979 = 39323#/2310 - 490362 20.06     Michiel Jansen 2012
10 571948 PRP12411 = 19*28751#/30 - 295468 20.01     Dana Jacobsen 2016
11 566040 PRP10449 = 24137#/2310 - 311774 23.53     Michiel Jansen 2012
12 538328 PRP10699 = 24821#/2310 - 362006 21.85     Michiel Jansen 2012
13 535836 PRP10569 = 24469#/30 - 374362 22.02     Dana Jacobsen 2015
14 440020 PRP9527 = 19*22091#/30 - 297762 20.06     Dana Jacobsen 2015
15 418250 PRP7713 = 11*17923#/46410 - 156344 23.55     Robert Smith 2015
16 384902 PRP7473 = 17383#/30 - 77702 22.37     Michiel Jansen 2012
17 383796 P7183 = 9527*(16673#)/2310 - 175622 23.21     Pierre Cami 2010
18 379304 PRP6261 = 79*14593#/46410 - 129600 26.31     Robert Smith 2015
19 360072 PRP7790 = 29*18089#/46410 - 190054 20.08     Robert Smith 2015
20 358374 PRP7581 = 17599#/2310 - 116826 20.53     Michiel Jansen 2012
Largest gap with basic expression:
-- 25146 P482 = 2^1600 + 248054360485 22.67     Dana Jacobsen 2014

 

Gaps which are the largest with at least that merit
Size Gap start Merit Discoverer Year
5103138 PRP216849 = 281*499979#/46410 - 2702372 10.22     Robert Smith 2016
4680156 PRP99750 = 230077#/2229464046810 - 3131794 20.38     Martin Raab 2016
1286500 PRP21608 = 1111111111111111111*49999#/(510510*499) - 525318 25.86     Hans Rosenthal 2016
1113106 P18662 = 587*43103#/2310 - 455704 25.90     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
379304 PRP6261 = 79*14593#/46410 - 129600 26.31     Robert Smith 2015
309030 PRP4223 = 1111111111111111111*9787#/(7#*641) - 130308 31.78     Hans Rosenthal 2015
157178 PRP2145 = 422569*5009#/30 - 96332 31.83     Dana Jacobsen 2016
120664 PRP1622 = 4999*3803#/510510 - 71716 32.32     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
71046 P948 = 961*2267#/39669630 - 53540 32.56     Dana Jacobsen 2015
66520 P816 = 1931*1933#/7230 - 30244 35.42     Michiel Jansen 2012
26892 P321 = 59740589*757#/210 - 14302 36.42     Dana Jacobsen 2016
13692 P163 = 1037600971*383#/210 - 8776 36.59     Dana Jacobsen 2016
10716 P127 = 7910896513*283#/30 - 6480 36.86     Dana Jacobsen 2016

 

gaps20nicely.txt contains the above gaps in Thomas R. Nicely's notation, but with full decimal expansions (may not display completely in some browsers) for primes with no short expression.

The Top-20 Prime Gaps for all merits contains a long list of the 20 largest known prime gaps with merit above m, for all m.
gaps20allnicely.txt contains those gaps in Nicely's notation.

Verifying the top-20 gaps with merit above 10 takes a long time due to the size and amount of the numbers. See Top 20 prime gap verifications for information about verifications.

Links:
Thomas R. Nicely's First occurrence prime gaps: http://www.trnicely.net/gaps/gaplist.html
Chris Caldwell's Prime Pages, The Gaps Between Primes: http://primes.utm.edu/notes/gaps.html
Eric Weisstein's Mathworld, Prime Gaps: http://mathworld.wolfram.com/PrimeGaps.html
Tomás Oliveira e Silva's Gaps between consecutive primes: http://www.ieeta.pt/~tos/gaps.html
Wikipedia's Prime gap: http://en.wikipedia.org/wiki/Prime_gap
Jens Kruse Andersen's
     First known prime megagap: http://primerecords.dk/primegaps/megagap.htm
     Largest known prime gap: http://primerecords.dk/primegaps/megagap2.htm
     A proven prime gap of 337446: http://primerecords.dk/primegaps/gap337446.htm
     New largest known prime gap: http://primerecords.dk/primegaps/megagap3.htm
     A megagap with merit 25.9: http://primerecords.dk/primegaps/gap1113106.htm
Carlos Rivera's The Prime Puzzles & Problem Connection: Problem 46 . Holes and Crowds-I
William V. Wright's Cramer's conjecture: http://wvwright.net

This page is based on an original page by Paul Leyland using partially different notation.
The Top-20 Prime Gaps is now maintained by Jens Kruse Andersen, jens.k.a@get2net.dk   home
Last updated 25 August 2016