Largest Known Primes

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The Prime Pages

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The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

Definitions and Notes

At this site we maintain a list of the 5000 largest known primes (plus selected smller primes). It is easy to show that there are infinitely many primes (in fact there are approximately x/ln(x) primes less than x, see "How Many"). This means that each of the primes below are probably separated by many millions of other (curently unkown) primes.

Record Primes of this Type

rank prime digits who when comment

1 2^82589933 - 1 24862048 G16 Dec 2018 Mersenne 51??
2 2^77232917 - 1 23249425 G15 Jan 2018 Mersenne 50??
3 2^74207281 - 1 22338618 G14 Jan 2016 Mersenne 49??
4 2^57885161 - 1 17425170 G13 Feb 2013 Mersenne 48
5 2^43112609 - 1 12978189 G10 Aug 2008 Mersenne 47
6 2^42643801 - 1 12837064 G12 Jun 2009 Mersenne 46
7 2^37156667 - 1 11185272 G11 Sep 2008 Mersenne 45
8 2^32582657 - 1 9808358 G9 Sep 2006 Mersenne 44
9 10223 · 2^31172165 + 1 9383761 SB12 Nov 2016
10 2^30402457 - 1 9152052 G9 Dec 2005 Mersenne 43
11 2^25964951 - 1 7816230 G8 Feb 2005 Mersenne 42
12 2^24036583 - 1 7235733 G7 May 2004 Mersenne 41
13 2^20996011 - 1 6320430 G6 Nov 2003 Mersenne 40
14 1059094^1048576 + 1 6317602 L4720 Nov 2018 Generalized Fermat
15 919444^1048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat
16 168451 · 2^19375200 + 1 5832522 L4676 Sep 2017
17 7 · 2^18233956 + 1 5488969 L4965 Oct 2020 Divides Fermat F(18233954)
18 3 · 2^17748034 - 1 5342692 L5404 Sep 2021
19 Phi(3, - 123447⁵²⁴²⁸⁸) 5338805 L4561 Feb 2017 Generalized unique
20 7 · 6^6772401 + 1 5269954 L4965 Sep 2019

References

Ribenboim95
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
Riesel94
H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics Vol, 126, Birkhäuser Boston, Boston, MA, 1994. ISBN 0-8176-3743-5. MR 95h:11142 [An excellent reference for those who want to start to program some of these algorithms. Code is provided in Pascal. Previous edition was vol. 57, 1985.]