The Largest Known Primes

Introduction (What are primes? Who cares?)
The Top Ten Record Primes:
largest, twin, Mersenne, primorial&factorial, and Sophie Germain
The Complete List of the Largest Known Primes
Other Sources of Prime Information
Euclid's Proof of the Infinitude of Primes
Comments? Suggestions? New records? New Links?

Primes: [ Home || Largest | Proving | How Many? | Mersenne | Glossary | Guestbook ]

Note: The correct URL for this page is http://www.utm.edu/research/primes/largest.html.

1. Introduction

An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. For example, the prime divisors of 10 are 2 and 5; and the first six primes are 2, 3, 5, 7, 11 and 13 (the first 10,000, and other lists are available). The Fundamental Theorem of Arithmetic shows that the primes are the building blocks of the positive integers: every positive integer is a product of prime numbers in one and only one way, except for the order of the factors.

The ancient Greeks proved (ca 300 BC) that there were infinitely many primes and that they were irregularly spaced (there can be arbitrarily large gaps between successive primes). On the other hand, in the nineteenth century it was shown that the number of primes less than or equal to n approaches n/(log n) (as n gets very large); so a rough estimate for the nth prime is n log n (see the document "How many primes are there?")

The Sieve of Eratosthenes is still the most efficient way of finding all very small primes (e.g., those less than 1,000,000). However, most of the largest primes are found using special cases of Lagrange's Theorem from group theory. See the separate documents on proving primality for more information.

In 1984 Samuel Yates defined a titanic prime to be any prime with at least 1,000 digits [Yates85]. When he introduced this term there were only 110 such primes known; now there are over 1000 times that many! And as computers and cryptology continually give new emphasis to search for ever larger primes, this number will continue to grow. Before long we expect to see the first megaprime.

At this site we maintain a database of the 5000 largest known primes (plus selected smaller primes). The complete list of approximately 6,000 primes is available in several forms below; but first we offer a quote and a few records for your perusal.

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. (Karl Friedrich Gauss, Disquisitiones Arithmeticae, 1801)

2. The "Top Ten" Record Primes

The Ten Largest Known Primes See also the page: The top 20: largest known primes.

On 27 January 1998, the team of Roland Clarkson, George Woltman, Scott Kurowski et. al. discovered a new record prime: 2^3021377-1. This is the 37th known Mersenne prime (there may be smaller ones as not all previous exponents have been checked). Clarkson found this prime using a program written by Woltman linked to the GIMPS internet database via Scott Kurowski's PrimeNet. Clarkson, a 19 year old college student, is one of about 4000 individuals involved in GIMPS: the Great Internet Mersenne Prime Search launched by Woltman in early 1996. GIMPS offers free software (and source code) for personal computer owners to use in searching for big prime numbers.

The primality of this number was verified by David Slowinski who has found several of the recent record primes. The complete decimal expansion of this 909,526 digit number is available in both text form and zipped form. Click here for more information.

prime	digits	who	when	reference
2^3021377-1	909526	Clarkson, Woltman, Kurowski & GIMPS	1998	(notes)
2^2976221-1	895932	Spence, Woltman & GIMPS	1997	(notes)
2^1398269-1	420921	Armengaud, Woltman & GIMPS	1996	(notes)
2^1257787-1	378632	Slowinski & Gage	1996	(notes)
2⁸⁵⁹⁴³³-1	258716	Slowinski & Gage	1994
2⁷⁵⁶⁸³⁹-1	227832	Slowinski & Gage	1992	[Peterson92]
302627325^.2⁵³⁰¹⁰¹+1	159585	Nash, Dunaieff, Burrowes, Jobling & Gallot	1999
481899^.2⁴⁸¹⁸⁹⁹+1	145072	Morii & Gallot	1998
361275^.2³⁶¹²⁷⁵+1	108761	Smith & Gallot	1998
302442855^.2³³⁶²¹¹+1	101219	Nash, Dunaieff, Burrowes, Jobling & Gallot	1998

Click here to see the one hundred largest known primes. You might also be interested in seeing this prime on the graph of the largest known prime by year.

The Ten Largest Known Twin Primes See also the page: The top 20: twin primes,
and the glossary entry: twin primes.

Twin primes are primes of the form p and p+2, i.e., they differ by two. It is conjectured, but not yet proven, that there are infinitely many twin primes (the same is true for all of the following forms of primes). Because discovering a twin prime actually involves finding two primes, the largest known twin primes are substantially smaller than the largest known primes of most other forms.

prime	digits	who	when	reference
361700055^.2³⁹⁰²⁰±1	11755	Henri Lifchitz	1999
835335^.2³⁹⁰¹⁴±1	11751	Ballinger & Gallot	1998
242206083^.2³⁸⁸⁸⁰±1	11713	Járai & Indlekofer	1995
40883037^.2²³⁴⁵⁶±1	7069	Lifchitz & Gallot	1998
843753^.2²²²²²±1	6696	Rivera & Gallot	1997
7485^.2²⁰⁰²³±1	6032	Buddenhagen & Gallot	1998
8182815^.2¹⁷⁸³⁸±1	5377	Smith & Gallot	1998
570918348^.10⁵¹²⁰±1	5129	Harvey Dubner	1995	[Ribenboim95, p263]
697053813^.2¹⁶³⁵²±1	4932	Járai & Indlekofer	1995	[IJ96]
37442007^.2¹⁵⁴⁴⁰±1	4656	Hanson & Gallot	1999

Click here to see all of the twin primes on the list of the Largest Known Primes.

Note: The idea of prime twins can be generalized to prime triplets, quadruplets; and more generally, prime k-tuplets. Tony Forbes keeps a page listing these records.

The Ten Largest Known Mersenne Primes See also the pages: The top 20: Mersenne primes,
and Mersenne primes (history, theorems and lists).

Mersenne primes are primes of the form 2^p-1. These are the easiest type of number to check for primality on a binary computer so they usually are also the largest primes known. Altogether 37 of these primes are known, but since the region between the largest two and the previous primes has not been completely searched we do not know if the largest is 37th according to size.

prime	digits	who	when	reference
2^3021377-1	909526	Clarkson, Woltman, Kurowski & GIMPS	1998	(notes)
2^2976221-1	895932	Spence, Woltman & GIMPS	1997	(notes)
2^1398269-1	420921	Armengaud, Woltman & GIMPS	1996	(notes)
2^1257787-1	378632	Slowinski & Gage	1996	(notes)
2⁸⁵⁹⁴³³-1	258716	Slowinski & Gage	1994
2⁷⁵⁶⁸³⁹-1	227832	Slowinski & Gage	1992	[Peterson92]
2²¹⁶⁰⁹¹-1	65050	David Slowinski	1985
2¹³²⁰⁴⁹-1	39751	David Slowinski	1983
2¹¹⁰⁵⁰³-1	33265	Welsh & Colquitt	1988	[CW91]
2⁸⁶²⁴³-1	25962	David Slowinski	1982	[Ewing83]

See our page on Mersenne numbers for more information including a complete table of the known Mersennes. You can also help fill in the gap by joining the Great Internet Mersenne Prime Search.

The Ten Largest Known Factorial/Primorial Primes See also: The top 20: primorial/factorial primes,
and the glossary entries: primorial, factorial.

Euclid's proof that there are infinitely many primes uses numbers of the form n#+1. Kummer's proof uses those of the form n#-1. Sometime students look at these proofs and assume the numbers n#+/-1 are always prime, but that is not so. When numbers of the form n#+/-1 are prime they are called primorial primes. Similarly numbers of the form n!+/-1 are called factorial primes. The current record holders and their discoverers are:

prime	digits	who	when	reference
6917!-1	23560	Caldwell & Gallot	1998
6380!+1	21507	Caldwell & Gallot	1998
42209#+1	18241	Caldwell & PrimeForm	1999
14614!!!!+1	13632	Charles F. Kerchner III	1998
10830!!!+1	13000	Charles F. Kerchner III	1998
3610!-1	11277	Chris Caldwell	1993	[Caldwell95]
3507!-1	10912	Chris Caldwell	1992	[Caldwell95]
24029#+1	10387	Chris Caldwell	1993	[Caldwell95]
23801#+1	10273	Chris Caldwell	1993	[Caldwell95]
11915!!!!!+1	8681	Charles F. Kerchner III	1998

Click here to see all of the known primorial, factorial and multifactorial primes on the list of the largest known primes.

The Ten Largest Known Sophie Germain Primes See also the page: The top 20: Sophie Germain,
and the glossary entry: Sophie Germain Prime.

A Sophie Germain prime is an odd prime p for which 2p+1 is also a prime. These were named after Sophie Germain when she proved that the first case of Fermat's Last Theorem (xⁿ+yⁿ=zⁿ has no solutions in non-zero integers for n>2) for exponents divisible by such primes. (Fermat's Last theorem has now been proved completely.)

prime	digits	who	when
18458709^.2³²⁶¹¹-1	9825	Kerchner & Gallot	1999
14516877^.2²⁴¹⁷⁶-1	7285	Kerchner & Gallot	1999
72021^.2²³⁶³⁰-1	7119	Yves Gallot	1998
2375063906985^.2¹⁹³⁸⁰-1	5847	Járai & Indlekofer	1999
276311^.2¹⁹⁰⁰³+1	5726	Ballinger & Gallot	1998
92305^.2¹⁶⁹⁹⁸+1	5122	Kerchner & Gallot	1998
8069496435^.10⁵⁰⁷²-1	5082	Harvey Dubner	1995
470943129^.2¹⁶³⁵²-1	4932	Járai & Indlekofer	1995
157324389^.2¹⁶³⁵²-1	4931	Járai & Indlekofer	1995
5415312903^.10⁴⁵²⁶-1	4536	Harvey Dubner	1994

Click here to see all of the Sophie Germain primes on the list of Largest Known Primes.

3. The Complete List of the Largest Known Primes

The current list of Largest Known Primes which contains the 5000 largest known primes, and other interesting smaller primes (with one thousand or more digits) is available in several ways:

As a searchable database
You may search the list by keyword, number size, discoverer...
all.txt
The whole list! This is a large file: 340K.
all.zip
The whole list (all.txt) pkZipped, so it is roughly one fourth the size of all.txt: 79K.
short.txt
All of the primes with 20,000 digits or more, plus the 'interesting' smaller primes (that is, those with comments on the list). So this is a much smaller file: 128K.

These files were last updated: Thursday, 10-Jun-99 13:14:46 CDT.

See also the other available lists of primes.

4. Other Sources of Large Primes

Because of the lag time between writing and printing, books can never keep up with the current prime records (so I created this page), but books can provide the mathematical theory behind these records much better than a limited series of web pages can. Recently there have been quite a number of excellent books published on primes and primality proving. Here are some of my favorite:

Paulo Ribenboim, "The New Book of Prime Number Records," Springer-Verlag, New York, 1995 (QA246 .R472) [Ribenboim95].
Paulo Ribenboim, "The Little Book of Big Primes," Springer-Verlag, 1991 (QA246 .R47) [Ribenboim91]. (A less mathematical version of the above text.)
Hans Riesel, "Prime Numbers and Computer Methods for Factorization," Progress in Mathematics, Birkhäuser Boston, vol. 126, 1994 [Riesel94].

See also [Bressoud89] and [Cohen93] on the page of partially annotated prime references. Also of interest is the Cunningham Project, an effort to factor the numbers in the title of the following book.

J. Brillhart, et al., "Factorizations of bⁿ±1, b = 2,3,5,6,7,10,11,12 up to high powers," American Mathematical Society, 1988 [BLSTW88].

The data from this project is available by web or by FTP.

See The Prime Page for many other web sources.

7. Comments, Suggestions? New records?

In order to help me maintain these lists please let me know of any corrections, comments, suggestions, flames, related WWW links and especially any new titanic primes!

Another prime page by Chris K. Caldwell <caldwell@utm.edu>