Largest Known Primes

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 282589933 - 1 24862048 G16 Dec 2018 Mersenne 51?? 2 277232917 - 1 23249425 G15 Jan 2018 Mersenne 50?? 3 274207281 - 1 22338618 G14 Jan 2016 Mersenne 49?? 4 257885161 - 1 17425170 G13 Feb 2013 Mersenne 48 5 243112609 - 1 12978189 G10 Aug 2008 Mersenne 47 6 242643801 - 1 12837064 G12 Jun 2009 Mersenne 46 7 Phi(3, - 4658591048576) 11887192 L4561 May 2023   8 237156667 - 1 11185272 G11 Sep 2008 Mersenne 45 9 232582657 - 1 9808358 G9 Sep 2006 Mersenne 44 10 10223 · 231172165 + 1 9383761 SB12 Nov 2016   11 230402457 - 1 9152052 G9 Dec 2005 Mersenne 43 12 225964951 - 1 7816230 G8 Feb 2005 Mersenne 42 13 224036583 - 1 7235733 G7 May 2004 Mersenne 41 14 19637361048576 + 1 6598776 L4245 Sep 2022 Generalized Fermat 15 19517341048576 + 1 6595985 L5583 Aug 2022 Generalized Fermat 16 202705 · 221320516 + 1 6418121 L5181 Dec 2021   17 220996011 - 1 6320430 G6 Nov 2003 Mersenne 40 18 10590941048576 + 1 6317602 L4720 Nov 2018 Generalized Fermat 19 9194441048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat 20 7 · 220267500 + 1 6101127 L4965 Jul 2022 Divides GF(20267499, 12) [GG]

Related Pages

• The Largest Known Primes
• Largest Known Prime by Year
• The chronology of prime number records

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.]