Prime Gaps
References:
Mainpage Prime Gap Project
& Mainthread on mersenneforum.org
Note:
Original page was created by Jens Kruse Andersen (2005 - 2019).
This page is developed and maintained by Norman Luhn (since Oct 2021).
Contact: pzktupel[at]pzktupel[dot]de.
Record tables
Largest known prime gaps
• more than 1 million
• between 900000 to 999999
• between 800000 to 899999
• between 700000 to 799999
• between 600000 to 699999
• between 500000 to 599999
• between 400000 to 499999
• between 300000 to 399999
Largest known prime gaps with proven primes as endpoints
Largest known prime gaps with merit 40..50
Top-1000 largest known prime gaps with merit 30..40
Top-1000 largest known prime gaps with merit 20..30
Top-1000 largest known prime gaps with merit 10..20
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 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.
This page is based on an original page by Paul Leyland
using partially different notation.