The Largest Known Primes--A Summary(A historic Prime Page resource
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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)See the FAQ for more infrmation on why we collect these large primes!
The Ten Largest Known Primes | See also the page: The top 20: largest known 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 2^{57885161}-1 17425170 G13 2013 Mersenne 48?? 2 2^{43112609}-1 12978189 G10 2008 Mersenne 47?? 3 2^{42643801}-1 12837064 G12 2009 Mersenne 46?? 4 2^{37156667}-1 11185272 G11 2008 Mersenne 45? 5 2^{32582657}-1 9808358 G9 2006 Mersenne 44? 6 2^{30402457}-1 9152052 G9 2005 Mersenne 43? 7 2^{25964951}-1 7816230 G8 2005 Mersenne 42 8 2^{24036583}-1 7235733 G7 2004 Mersenne 41 9 2^{20996011}-1 6320430 G6 2003 Mersenne 40 10 2^{13466917}-1 4053946 G5 2001 Mersenne 39 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.
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.
rank prime digits who when reference 1 3756801695685·2^{666669}+1 200700 L1921 2011 Twin (p+2) 2 3756801695685·2^{666669}-1 200700 L1921 2011 Twin (p) 3 65516468355·2^{333333}+1 100355 L923 2009 Twin (p+2) 4 65516468355·2^{333333}-1 100355 L923 2009 Twin (p) 5 2003663613·2^{195000}+1 58711 L202 2007 Twin (p+2) 6 2003663613·2^{195000}-1 58711 L202 2007 Twin (p) 7 38529154785·2^{173250}+1 52165 L3494 2014 Twin (p+2) 8 38529154785·2^{173250}-1 52165 L3494 2014 Twin (p) 9 194772106074315·2^{171960}+1 51780 x24 2007 Twin (p+2) 10 194772106074315·2^{171960}-1 51780 x24 2007 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.
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. GIMPS is steadily finding these behemoths!
rank prime digits who when reference 1 2^{57885161}-1 17425170 G13 2013 Mersenne 48?? 2 2^{43112609}-1 12978189 G10 2008 Mersenne 47?? 3 2^{42643801}-1 12837064 G12 2009 Mersenne 46?? 4 2^{37156667}-1 11185272 G11 2008 Mersenne 45? 5 2^{32582657}-1 9808358 G9 2006 Mersenne 44? 6 2^{30402457}-1 9152052 G9 2005 Mersenne 43? 7 2^{25964951}-1 7816230 G8 2005 Mersenne 42 8 2^{24036583}-1 7235733 G7 2004 Mersenne 41 9 2^{20996011}-1 6320430 G6 2003 Mersenne 40 10 2^{13466917}-1 4053946 G5 2001 Mersenne 39 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
and 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. 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.
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 (x^{n}+y^{n}=z^{n} 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 18543637900515·2^{666667}-1 200701 L2429 2012 Sophie Germain (p) 2 183027·2^{265440}-1 79911 L983 2010 Sophie Germain (p) 3 648621027630345·2^{253824}-1 76424 x24 2009 Sophie Germain (p) 4 620366307356565·2^{253824}-1 76424 x24 2009 Sophie Germain (p) 5 607095·2^{176311}-1 53081 L983 2009 Sophie Germain (p) 6 48047305725·2^{172403}-1 51910 L99 2007 Sophie Germain (p) 7 137211941292195·2^{171960}-1 51780 x24 2006 Sophie Germain (p) 8 31737014565·2^{140003}-1 42156 L95 2010 Sophie Germain (p) 9 14962863771·2^{140001}-1 42155 L95 2010 Sophie Germain (p) 10 33759183·2^{123458}-1 37173 L527 2009 Sophie Germain (p) Click here to see all of the Sophie Germain primes on the list of Largest Known Primes.