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 66520 P816 = 1931*1933#/7230 - 30244 35.4245 Michiel Jansen 2012
2 1476 P19 = 1425172824437699411 35.3103 Tomás Oliveira e Silva 2009
3 1442 P18 = 804212830686677669 34.9757 Siegfried Herzog & Tomás Oliveira e Silva 2005
4 1550 P20 = 18361375334787046697 34.9439 Bertil Nyman 2014
5 1530 P20 = 17678654157568189057 34.5225 Bertil Nyman 2014
6 1454 P19 = 3219107182492871783 34.1189 Silvio Pardi & Tomás Oliveira e Silva 2011
7 1370 P18 = 418032645936712127 33.7652 Donald E. Knuth 2006
8 1490 P20 = 17849040361018364489 33.6127 Bertil Nyman 2011
9 1440 P19 = 4253027105513399527 33.5710 Leif Leonhardy 2014
10 18508 P241 = 227939*587#/11370 - 6508 33.4563 Dana Jacobsen 2014
11 1356 P18 = 401429925999153707 33.4536 Donald E. Knuth 2006
12 1358 P18 = 523255220614645319 33.2853 Siegfried Herzog & Tomás Oliveira e Silva 2007
13 1476 P20 = 18227591035187773493 33.2811 Bertil Nyman 2013
14 1380 P19 = 1031501833130243273 33.2710 Siegfried Herzog & Tomás Oliveira e Silva 2007
15 1392 P19 = 1480032037939634731 33.2707 Tomás Oliveira e Silva 2009
16 1410 P19 = 2635281932481539903 33.2425 Siegfried Herzog & Tomás Oliveira e Silva 2011
17 1418 P19 = 3725235533504101511 33.1605 Tomás Oliveira e Silva 2012
18 1416 P19 = 3750992529339978877 33.1084 Tomás Oliveira e Silva 2012
19 2412 P32 = 51365308007314298022939444480671 33.0336 Helmut Spielauer 2011
20 1398 P19 = 2424708729726767749 33.0245 Tomás Oliveira e Silva 2011

 

Top-20 gaps with merit above 10
Rank Size Gap start Merit Discoverer Year
1 3311852 PRP97953 = 226007#/2310 - 2305218 14.68     Michiel Jansen & Jens K. Andersen 2012
2 2765878 PRP100006 = 230567#/2310 - 939244 12.01     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
3 2724214 PRP100000 = 230563#/2310 - 44352 11.83     Michiel Jansen & Jens K. Andersen 2013
4 2254930 PRP86853 = 1122483511... 11.28     Hans Rosenthal & Jens K. Andersen 2004
5 2055816 PRP56962 = 6887*(131591#)/2730 - 1381994 15.67     Pierre Cami 2010
6 1575828 PRP45334 = 104729#/2310 - 1282742 15.10     Michiel Jansen 2012
Largest gap with proven end points:
7 1113106 P18662 = 587*43103#/2310 - 455704 25.90     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
8 1078180 PRP38007 = 50491*(87811#)/6 - 657714 12.32     Pierre Cami 2006
9 1001548 PRP43429 = 1913094464... 10.02      Hans Rosenthal & Jens K. Andersen 2004
10 984108 PRP16901 = 39161#/2310 - 510478 25.29     Michiel Jansen 2012
11 973764 PRP18648 = 431*43063#/2310 - 278398 22.68     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
12 875274 PRP19263 = 44579#/2310 - 379448 19.73     Pierre Cami 2013
13 865056 PRP18636 = 44633*(43037#)/2310 - 394442 20.16     Pierre Cami 2013
14 856608 PRP19621 = 45413#/2310 - 459642 18.96     Pierre Cami 2013
15 843888 PRP18636 = 44753*(43037#)/2310 - 177806 19.67     Pierre Cami 2013
16 834114 PRP16497 = 38231#/2310 - 393706 21.96     Michiel Jansen 2012
17 830202 PRP18636 = 44351*(43037#)/2310 - 549476 19.35     Pierre Cami 2013
18 826800 PRP18657 = 587*43093#/2310 - 365798 19.25     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
19 825658 PRP29317 = 51307*(67777#)/6 - 327676 12.23     Pierre Cami 2006
20 818886 PRP17071 = 39541#/2310 - 380216 20.83     Michiel Jansen 2012
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 1113106 P18662 = 587*43103#/2310 - 455704 25.90     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
2 984108 PRP16901 = 39161#/2310 - 510478 25.29     Michiel Jansen 2012
3 973764 PRP18648 = 431*43063#/2310 - 278398 22.68     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
4 865056 PRP18636 = 44633*(43037#)/2310 - 394442 20.16     Pierre Cami 2013
5 834114 PRP16497 = 38231#/2310 - 393706 21.96     Michiel Jansen 2012
6 818886 PRP17071 = 39541#/2310 - 380216 20.83     Michiel Jansen 2012
7 784246 PRP16979 = 39323#/2310 - 490362 20.06     Michiel Jansen 2012
8 566040 PRP10449 = 24137#/2310 - 311774 23.53     Michiel Jansen 2012
9 538328 PRP10699 = 24821#/2310 - 362006 21.85     Michiel Jansen 2012
10 384902 PRP7473 = 17383#/30 - 77702 22.37     Michiel Jansen 2012
11 383796 P7183 = 9527*(16673#)/2310 - 175622 23.21     Pierre Cami 2010
12 358374 PRP7581 = 17599#/2310 - 116826 20.53     Michiel Jansen 2012
13 326404 PRP6797 = 15761#/210 - 127858 20.86     Michiel Jansen 2012
14 324618 PRP6319 = 14731#/2310 - 206846 22.31     Michiel Jansen 2012
15 318876 PRP6800 = 15767#/2310 - 171714 20.37     Michiel Jansen 2012
16 304950 PRP6447 = 19*14983#/30 - 91346 20.54     Dana Jacobsen 2014
17 303276 PRP6095 = 14197#/210 - 76898 21.61     Michiel Jansen 2011
18 296318 PRP6110 = 14249#/30030 - 74924 21.06     Michiel Jansen 2012
19 294642 PRP6373 = 14827#/2310 - 161206 20.08     Michiel Jansen 2012
20 289534 PRP5688 = 7*13217#/30 - 176068 22.11     Dana Jacobsen 2014
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
3311852 PRP97953 = 226007#/2310 - 2305218 14.68     Michiel Jansen & Jens K. Andersen 2012
2055816 PRP56962 = 6887*(131591#)/2730 - 1381994 15.67     Pierre Cami 2010
1113106 P18662 = 587*43103#/2310 - 455704 25.90     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
211044 PRP3429 = 41*8011#/30 - 164938 26.74     Dana Jacobsen 2014
143692 PRP2243 = 6091*5261#/210 - 51838 27.83     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
128130 PRP1992 = 4759*4657#/30 - 83388 27.94     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
120664 PRP1622 = 4999*3803#/510510 - 71716 32.32     Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
66520 P816 = 1931*1933#/7230 - 30244 35.42     Michiel Jansen 2012

 

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 11 September 2014