The Largest Known Primes--A Summary (A historic Prime Page resource since 1994!) Last modified: 05:20:23 AM Monday February 1 2016 UTC New record prime: 274,207,281-1 with 22,338,618 digits by Cooper, Woltman, Kurowski, Blosser & GIMPS (7 Jan 2016).

Contents:

1. Introduction (What are primes? Who cares?)
2. The Top Ten Record Primes:
       largest, twin, Mersenne, primorial & factorial, and Sophie Germain
3. The Complete List of the Largest Known Primes
4. Other Sources of Prime Information
5. Euclid's Proof of the Infinitude of Primes

  1. Introduction

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 [Yates84, 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 twenty-five million digit prime.

If you want to understand a building, how it will react to weather or fire, you first need to know what it is made of. The same is true for the integers--most of their properties can be traced back to what they are made of: their prime factors. For example, in Euclid's Geometry (over 2,000 years ago), Euclid studied even perfect numbers and traced them back to what we now call Mersenne primes.

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. (Carl Friedrich Gauss, Disquisitiones Arithmeticae, 1801)

  2. The "Top Ten" Record Primes

The largest known prime has almost always been a Mersenne prime.  Why Mersennes?  Because the way the largest numbers N are proven prime is based on the factorizations of either N+1 or N-1, and for Mersennes the factorization of N+1 is as trivial as possible (a power of two). 

The Great Internet Mersenne Prime Search (GIMPS) was launched by George Woltman in early 1996, and has had a virtual lock on the largest known prime since then.  This is because its excellent free software is easy to install and maintain, requiring little of the user other than watch and see if they find the next big one and maybe win some EFF prize money!

Any record in this list of the top ten is a testament to the incredible amount of work put in by the programmers, project directors (GIMPS, Seventeen or Bust, Generalized Fermat Search...), and the tens of thousands of enthusiasts!

rank	prime	digits	who	when	reference
1	274207281-1	22338618	G14	2016	Mersenne 49??
2	257885161-1	17425170	G13	2013	Mersenne 48?
3	243112609-1	12978189	G10	2008	Mersenne 47?
4	242643801-1	12837064	G12	2009	Mersenne 46?
5	237156667-1	11185272	G11	2008	Mersenne 45?
6	232582657-1	9808358	G9	2006	Mersenne 44
7	230402457-1	9152052	G9	2005	Mersenne 43
8	225964951-1	7816230	G8	2005	Mersenne 42
9	224036583-1	7235733	G7	2004	Mersenne 41
10	220996011-1	6320430	G6	2003	Mersenne 40

Click here to see the one hundred largest known primes. You might also be interested in seeing the graph of the size of record primes by year: throughout history or just in the last decade.

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.

rank	prime	digits	who	when	reference
1	3756801695685·2666669+1	200700	L1921	2011	Twin (p+2)
2	3756801695685·2666669-1	200700	L1921	2011	Twin (p)
3	65516468355·2333333+1	100355	L923	2009	Twin (p+2)
4	65516468355·2333333-1	100355	L923	2009	Twin (p)
5	4884940623·2198800+1	59855	L4166	2015	Twin (p+2)
6	4884940623·2198800-1	59855	L4166	2015	Twin (p)
7	2003663613·2195000+1	58711	L202	2007	Twin (p+2)
8	2003663613·2195000-1	58711	L202	2007	Twin (p)
9	38529154785·2173250+1	52165	L3494	2014	Twin (p+2)
10	38529154785·2173250-1	52165	L3494	2014	Twin (p)

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.

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.  GIMPS is steadily finding these behemoths!

rank	prime	digits	who	when	reference
1	274207281-1	22338618	G14	2016	Mersenne 49??
2	257885161-1	17425170	G13	2013	Mersenne 48?
3	243112609-1	12978189	G10	2008	Mersenne 47?
4	242643801-1	12837064	G12	2009	Mersenne 46?
5	237156667-1	11185272	G11	2008	Mersenne 45?
6	232582657-1	9808358	G9	2006	Mersenne 44
7	230402457-1	9152052	G9	2005	Mersenne 43
8	225964951-1	7816230	G8	2005	Mersenne 42
9	224036583-1	7235733	G7	2004	Mersenne 41
10	220996011-1	6320430	G6	2003	Mersenne 40

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.

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.  Sometimes 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:

rank	prime	digits	who	when	reference
1	1098133#-1	476311	p346	2012	Primorial
2	843301#-1	365851	p302	2010	Primorial
3	392113#+1	169966	p16	2001	Primorial
4	366439#+1	158936	p16	2001	Primorial
5	145823#+1	63142	p21	2000	Primorial
6	42209#+1	18241	p8	1999	Primorial
7	24029#+1	10387	C	1993	Primorial
8	23801#+1	10273	C	1993	Primorial
9	18523#+1	8002	D	1989	Primorial
10	15877#-1	6845	CD	1992	Primorial

rank	prime	digits	who	when	reference
1	150209!+1	712355	p3	2011	Factorial
2	147855!-1	700177	p362	2013	Factorial
3	110059!+1	507082	p312	2011	Factorial
4	103040!-1	471794	p301	2010	Factorial
5	94550!-1	429390	p290	2010	Factorial
6	34790!-1	142891	p85	2002	Factorial
7	26951!+1	107707	p65	2002	Factorial
8	21480!-1	83727	p65	2001	Factorial
9	6917!-1	23560	g1	1998	Factorial
10	6380!+1	21507	g1	1998	Factorial

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

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 by Andrew Wiles.

rank	prime	digits	who	when	reference
1	18543637900515·2666667-1	200701	L2429	2012	Sophie Germain (p)
2	183027·2265440-1	79911	L983	2010	Sophie Germain (p)
3	648621027630345·2253824-1	76424	x24	2009	Sophie Germain (p)
4	620366307356565·2253824-1	76424	x24	2009	Sophie Germain (p)
5	607095·2176311-1	53081	L983	2009	Sophie Germain (p)
6	48047305725·2172403-1	51910	L99	2007	Sophie Germain (p)
7	137211941292195·2171960-1	51780	x24	2006	Sophie Germain (p)
8	31737014565·2140003-1	42156	L95	2010	Sophie Germain (p)
9	14962863771·2140001-1	42155	L95	2010	Sophie Germain (p)
10	33759183·2123458-1	37173	L527	2009	Sophie Germain (p)

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

  3. Other Sources of Large Primes

• P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, 1995. (QA246 .R472).
• P. Ribenboim, The little book of bigger primes, Springer-Verlag, New York, 2004.  (A less mathematical version of the above text.)
• H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics volume 126, Birkäuser Boston, 1994.
• R. Crandall and C. Pomerance, Prime numbers: a computational perspective, Springer-Verlag, New York, 2001.  ISBN 0-387-94777-9.

• 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].