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.

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. 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.

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
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 November 2009