## 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. (This is the key to their importance: the prime factors of an integer determines its properties.) 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

*n*th 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 fifteen milliondigit 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. (See the FAQ for more infrmation on why we collect these large primes!Carl 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. |

The largest known prime has almost always been a Mersenne prime. Why Mersennes? Because the way the largest numbersNare proven prime is based on the factorizations of eitherN+1 orN-1, and for Mersennes the factorization ofN+1 is as trivial as possible (a power of two). TheGreatInternetMersennePrimeSearch (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! Tens of thousands of users have replaced the ubiquitous inane "screen savers" with this much more productive use of their computer's idle time (and the hope of winning 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!

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.

rank prime digits who when reference 1 2^{43112609}-112978189 G10 2008 Mersenne 47?? 2 2^{42643801}-112837064 G12 2009 Mersenne 46?? 3 2^{37156667}-111185272 G11 2008 Mersenne 45?? 4 2^{32582657}-19808358 G9 2006 Mersenne 44?? 5 2^{30402457}-19152052 G9 2005 Mersenne 43?? 6 2^{25964951}-17816230 G8 2005 Mersenne 42? 7 2^{24036583}-17235733 G7 2004 Mersenne 41? 8 2^{20996011}-16320430 G6 2003 Mersenne 40 9 2^{13466917}-14053946 G5 2001 Mersenne 39 10 19249·2^{13018586}+13918990 SB10 2007

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

Twin primesare primes of the formpandp+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.

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

rank prime digits who when reference 1 65516468355·2^{333333}+1100355 L923 2009 Twin (p+2) 2 65516468355·2^{333333}-1100355 L923 2009 Twin (p) 3 2003663613·2^{195000}+158711 L202 2007 Twin (p+2) 4 2003663613·2^{195000}-158711 L202 2007 Twin (p) 5 194772106074315·2^{171960}+151780 x24 2007 Twin (p+2) 6 194772106074315·2^{171960}-151780 x24 2007 Twin (p) 7 100314512544015·2^{171960}+151780 x24 2006 Twin (p+2) 8 100314512544015·2^{171960}-151780 x24 2006 Twin (p) 9 16869987339975·2^{171960}+151779 x24 2005 Twin (p+2) 10 16869987339975·2^{171960}-151779 x24 2005 Twin (p)

Note:The idea of prime twins can be generalized to prime triplets, quadruplets; and more generally, primek-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 primesare 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!

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.

rank prime digits who when reference 1 2^{43112609}-112978189 G10 2008 Mersenne 47?? 2 2^{42643801}-112837064 G12 2009 Mersenne 46?? 3 2^{37156667}-111185272 G11 2008 Mersenne 45?? 4 2^{32582657}-19808358 G9 2006 Mersenne 44?? 5 2^{30402457}-19152052 G9 2005 Mersenne 43?? 6 2^{25964951}-17816230 G8 2005 Mersenne 42? 7 2^{24036583}-17235733 G7 2004 Mersenne 41? 8 2^{20996011}-16320430 G6 2003 Mersenne 40 9 2^{13466917}-14053946 G5 2001 Mersenne 39 10 2^{6972593}-12098960 G4 1999 Mersenne 38

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 formn#+1. Kummer's proof uses those of the formn#-1. Sometimes students look at these proofs and assume the numbersn#+/-1 are always prime, but that is not so. When numbers of the formn#+/-1 are prime they are calledprimorial primes. Similarly numbers of the formn!+/-1 are calledfactorial primes. The current record holders and their discoverers are:

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

rank prime digits who when reference 1 843301#-1365851 p302 2010 Primorial 2 392113#+1169966 p16 2001 Primorial 3 366439#+1158936 p16 2001 Primorial 4 145823#+163142 p21 2000 Primorial 5 42209#+118241 p8 1999 Primorial 6 24029#+110387 C 1993 Primorial 7 23801#+110273 C 1993 Primorial 8 18523#+18002 D 1989 Primorial 9 15877#-16845 CD 1992 Primorial 10 13649#+15862 D 1987 Primorial

rank prime digits who when reference 1 103040!-1471794 p301 2010 Factorial 2 94550!-1429390 p290 2010 Factorial 3 34790!-1142891 p85 2002 Factorial 4 26951!+1107707 p65 2002 Factorial 5 21480!-183727 p65 2001 Factorial 6 6917!-123560 g1 1998 Factorial 7 6380!+121507 g1 1998 Factorial 8 3610!-111277 C 1993 Factorial 9 3507!-110912 C 1992 Factorial 10 1963!-15614 CD 1992 Factorial

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

*p*for which 2

*p*+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 | 183027·2^{265440}-1 | 79911 | L983 | 2010 | Sophie Germain (p) |

2 | 648621027630345·2^{253824}-1 | 76424 | x24 | 2009 | Sophie Germain (p) |

3 | 620366307356565·2^{253824}-1 | 76424 | x24 | 2009 | Sophie Germain (p) |

4 | 607095·2^{176311}-1 | 53081 | L983 | 2009 | Sophie Germain (p) |

5 | 48047305725·2^{172403}-1 | 51910 | L99 | 2007 | Sophie Germain (p) |

6 | 137211941292195·2^{171960}-1 | 51780 | x24 | 2006 | Sophie Germain (p) |

7 | 31737014565·2^{140003}-1 | 42156 | L95 | 2010 | Sophie Germain (p) |

8 | 14962863771·2^{140001}-1 | 42155 | L95 | 2010 | Sophie Germain (p) |

9 | 33759183·2^{123458}-1 | 37173 | L527 | 2009 | Sophie Germain (p) |

10 | 7068555·2^{121301}-1 | 36523 | L100 | 2005 | Sophie Germain (p) |

## No comments:

Post a Comment