Monday, January 17, 2011

The biggest prime number

[up]   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 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 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. (Carl Friedrich Gauss, Disquisitiones Arithmeticae, 1801)
See the FAQ for more infrmation on why we collect these large primes!

[up]   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 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!  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!

rankprime digitswhowhenreference
1243112609-1 12978189 G102008 Mersenne 47??
2242643801-1 12837064 G122009 Mersenne 46??
3237156667-1 11185272 G112008 Mersenne 45??
4232582657-1 9808358 G92006 Mersenne 44??
5230402457-1 9152052 G92005 Mersenne 43??
6225964951-1 7816230 G82005 Mersenne 42?
7224036583-1 7235733 G72004 Mersenne 41?
8220996011-1 6320430 G62003 Mersenne 40
9213466917-1 4053946 G52001 Mersenne 39
1019249·213018586+1 3918990 SB102007
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.
rankprime digitswhowhenreference
165516468355·2333333+1 100355 L9232009 Twin (p+2)
265516468355·2333333-1 100355 L9232009 Twin (p)
32003663613·2195000+1 58711 L2022007 Twin (p+2)
42003663613·2195000-1 58711 L2022007 Twin (p)
5194772106074315·2171960+1 51780 x242007 Twin (p+2)
6194772106074315·2171960-1 51780 x242007 Twin (p)
7100314512544015·2171960+1 51780 x242006 Twin (p+2)
8100314512544015·2171960-1 51780 x242006 Twin (p)
916869987339975·2171960+1 51779 x242005 Twin (p+2)
1016869987339975·2171960-1 51779 x242005 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 2p-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!
rankprime digitswhowhenreference
1243112609-1 12978189 G102008 Mersenne 47??
2242643801-1 12837064 G122009 Mersenne 46??
3237156667-1 11185272 G112008 Mersenne 45??
4232582657-1 9808358 G92006 Mersenne 44??
5230402457-1 9152052 G92005 Mersenne 43??
6225964951-1 7816230 G82005 Mersenne 42?
7224036583-1 7235733 G72004 Mersenne 41?
8220996011-1 6320430 G62003 Mersenne 40
9213466917-1 4053946 G52001 Mersenne 39
1026972593-1 2098960 G41999 Mersenne 38
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:
rankprime digitswhowhenreference
1843301#-1 365851 p3022010 Primorial
2392113#+1 169966 p162001 Primorial
3366439#+1 158936 p162001 Primorial
4145823#+1 63142 p212000 Primorial
542209#+1 18241 p81999 Primorial
624029#+1 10387 C1993 Primorial
723801#+1 10273 C1993 Primorial
818523#+1 8002 D1989 Primorial
915877#-1 6845 CD1992 Primorial
1013649#+1 5862 D1987 Primorial

rankprime digitswhowhenreference
1103040!-1 471794 p3012010 Factorial
294550!-1 429390 p2902010 Factorial
334790!-1 142891 p852002 Factorial
426951!+1 107707 p652002 Factorial
521480!-1 83727 p652001 Factorial
66917!-1 23560 g11998 Factorial
76380!+1 21507 g11998 Factorial
83610!-1 11277 C1993 Factorial
93507!-1 10912 C1992 Factorial
101963!-1 5614 CD1992 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 (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.
rankprime digitswhowhenreference
1183027·2265440-1 79911 L9832010 Sophie Germain (p)
2648621027630345·2253824-1 76424 x242009 Sophie Germain (p)
3620366307356565·2253824-1 76424 x242009 Sophie Germain (p)
4607095·2176311-1 53081 L9832009 Sophie Germain (p)
548047305725·2172403-1 51910 L992007 Sophie Germain (p)
6137211941292195·2171960-1 51780 x242006 Sophie Germain (p)
731737014565·2140003-1 42156 L952010 Sophie Germain (p)
814962863771·2140001-1 42155 L952010 Sophie Germain (p)
933759183·2123458-1 37173 L5272009 Sophie Germain (p)
107068555·2121301-1 36523 L1002005 Sophie Germain (p)
Click here to see all of the Sophie Germain primes on the list of Largest Known Primes

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