Largest Known Primes

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The Prime Pages

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.

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 202705 · 2^21320516 + 1 6418121 L5181 Dec 2021
14 2^20996011 - 1 6320430 G6 Nov 2003 Mersenne 40
15 1059094^1048576 + 1 6317602 L4720 Nov 2018 Generalized Fermat
16 919444^1048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat
17 168451 · 2^19375200 + 1 5832522 L4676 Sep 2017
18 69 · 2^18831865 - 1 5668959 L4965 Dec 2021
19 7 · 2^18233956 + 1 5488969 L4965 Oct 2020 Divides Fermat F(18233954)
20 3 · 2^18196595 - 1 5477722 L5461 Jan 2022

References

Ribenboim95
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, 1995. New York, NY, 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, 1994. Boston, MA, 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.]