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 p 1 and p 2 as end points, if p 1 < p 2 are consecutive primes (all intermediate numbers are composites). Some people define p 1 +1 and p 2 -1 to be the end points.

The size of the prime gap is p 2 - p 1 . Some people define it to be one less.

The merit of the prime gap is size / ln p 1 , where ln is the natural logarithm. Some people use p 2 or a number between p 1 and p 2 , 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.

Timothy D. Winslow

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 p 1 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 18306 P209 = 650094367*491#/2310 - 8936 38.07 Dana Jacobsen 2017 2 10716 P127 = 7910896513*283#/30 - 6480 36.86 Dana Jacobsen 2016 3 13692 P163 = 1037600971*383#/210 - 8776 36.59 Dana Jacobsen 2016 4 26892 P321 = 59740589*757#/210 - 14302 36.42 Dana Jacobsen 2016 5 11924 P147 = 4588394369*347#/30 - 7200 35.45 Dana Jacobsen 2016 6 66520 P816 = 1931*1933#/7230 - 30244 35.42 Michiel Jansen 2012 7 1476 P19 = 1425172824437699411 35.3103 Tomás Oliveira e Silva 2009 8 19474 P241 = 1485582109*571#/210 - 7576 35.20 Dana Jacobsen 2016 9 1442 P18 = 804212830686677669 34.9757 Siegfried Herzog & Tomás Oliveira e Silva 2005 10 24008 P299 = 171346649*701#/210 - 6918 34.96 Dana Jacobsen 2016 11 1550 P20 = 18361375334787046697 34.9439 Bertil Nyman 2014 12 10942 P137 = 6436615289*313#/30 - 1942 34.91 Dana Jacobsen 2016 13 18840 P236 = 962682899*563#/30 - 3918 34.76 Dana Jacobsen 2016 14 11350 P142 = 693236519*337#/210 - 2778 34.75 Dana Jacobsen 2016 15 19170 P241 = 820696571*571#/210 - 9428 34.69 Dana Jacobsen 2016 16 13704 P172 = 558020653*409#/2310 - 4196 34.68 Dana Jacobsen 2016 17 27666 P347 = 46349070720025037040361858900129... 34.66 Gapcoin 2017 18 21126 P266 = 844298401*619#/210 - 10510 34.62 Dana Jacobsen 2016 19 12176 P153 = 577924573*359#/30 - 7508 34.61 Dana Jacobsen 2016 20 22040 P278 = 1371215149*647#/30 - 9708 34.52 Dana Jacobsen 2017

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 18306 P209 = 650094367*491#/2310 - 8936 38.07 Dana Jacobsen 2017

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 19 January 2017