**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.

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 e^{m} prime gaps has merit > m.
e^{10} ~= 20000, e^{20} ~= 5·10^{8}, e^{30} ~= 10^{13}.
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.

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 | 1454 | P19 = 3219107182492871783 | 34.1189 | Silvio Pardi & Tomás Oliveira e Silva | 2011 |

5 | 1370 | P18 = 418032645936712127 | 33.7652 | Donald E. Knuth | 2006 |

6 | 1490 | P20 = 17849040361018364489 | 33.6127 | Bertil Nyman | 2011 |

7 | 1356 | P18 = 401429925999153707 | 33.4536 | Donald E. Knuth | 2006 |

8 | 1358 | P18 = 523255220614645319 | 33.2853 | Siegfried Herzog & Tomás Oliveira e Silva | 2007 |

9 | 1476 | P20 = 18227591035187773493 | 33.2811 | Bertil Nyman | 2013 |

10 | 1380 | P19 = 1031501833130243273 | 33.2710 | Siegfried Herzog & Tomás Oliveira e Silva | 2007 |

11 | 1392 | P19 = 1480032037939634731 | 33.2707 | Tomás Oliveira e Silva | 2009 |

12 | 1410 | P19 = 2635281932481539903 | 33.2425 | Siegfried Herzog & Tomás Oliveira e Silva | 2011 |

13 | 1418 | P19 = 3725235533504101511 | 33.1605 | Tomás Oliveira e Silva | 2012 |

14 | 1416 | P19 = 3750992529339978877 | 33.1084 | Tomás Oliveira e Silva | 2012 |

15 | 2412 | P32 = 51365308007314298022939444480671 | 33.0336 | Helmut Spielauer | 2011 |

16 | 1398 | P19 = 2424708729726767749 | 33.0245 | Tomás Oliveira e Silva | 2011 |

17 | 1364 | P19 = 1051140888051230423 | 32.8703 | Siegfried Herzog & Tomás Oliveira e Silva | 2007 |

18 | 1328 | P18 = 352521223451364323 | 32.8681 | Tomás Oliveira e Silva | 2006 |

19 | 1400 | P19 = 3431657795858378003 | 32.8026 | Silvio Pardi & Tomás Oliveira e Silva | 2012 |

20 | 1360 | P19 = 1153277647303540597 | 32.7008 | Siegfried Herzog & Tomás Oliveira e Silva | 2008 |

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 |

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 | 303276 | PRP6095 = 14197#/210 - 76898 | 21.61 | Michiel Jansen | 2011 |

17 | 296318 | PRP6110 = 14249#/30030 - 74924 | 21.06 | Michiel Jansen | 2012 |

18 | 294642 | PRP6373 = 14827#/2310 - 161206 | 20.08 | Michiel Jansen | 2012 |

19 | 283266 | PRP6037 = 14033#/330 - 210758 | 20.38 | Michiel Jansen | 2012 |

20 | 269234 | PRP4737 = 11057#/2310 - 196998 | 24.68 | Michiel Jansen | 2012 |

Largest gap with basic expression: | |||||

-- | 18510 | P362 = 2^1200 + 889723198107 | 22.25 | Hans Rosenthal & Jim Fougeron | 2004 |

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 |

201832 | P3379 = 14747*(7879#)/2310 - 65974 | 25.95 | Pierre Cami | 2010 |

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://users.cybercity.dk/~dsl522332/math/primegaps/megagap.htm

Largest known prime gap: http://users.cybercity.dk/~dsl522332/math/primegaps/megagap2.htm

A proven prime gap of 337446: http://users.cybercity.dk/~dsl522332/math/primegaps/gap337446.htm

New largest known prime gap: http://users.cybercity.dk/~dsl522332/math/primegaps/megagap3.htm

A megagap with merit 25.9: http://users.cybercity.dk/~dsl522332/math/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 23 February 2014