The Top Twenty--a Prime Page Collection

How big is big enough?

This page : Introduction | Table of minimal sizes | Sometimes more?
  View this page in:   language help

(up)Introduction

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. To make the top 5000 today a prime must have 388342 digits. This is increasing at roughly 10,000 digits per year. Click on the trends tab above to view the change over the last five years.

Smaller primes, those not large enough to be in the top 5000, may stay on the list if they are in the first few (either 5 or 20). Below we list how large they must be to make our list. But be careful, this is a moving target--every month the size of these records increase. So if you want to stay on the list for awhile, do not search for a prime with just a few digits more, aim for a few thousand digits more!

(up)Table of minimal sizes

Smallest prime of special forms on the list
(the smallest that make the list on the merit of the indicated comment alone)
digits required archivable form or class number archived number on list
(**) Arithmetic progression (1,d=*) (**) 10
(**) Arithmetic progression (2,d=*) (**) 11
388342 Arithmetic progression (3,d=*) 5 13
13556 Arithmetic progression (4,d=*) 5 10
7026 Arithmetic progression (5,d=*) 5 10
3019 Arithmetic progression (6,d=*) 5 5
1290 Arithmetic progression (7,d=*) 5 5
1014 Arithmetic progression (8,d=*) 5 5
1014 Arithmetic progression (9,d=*) 5 5
(**) Consecutive primes in arithmetic progression (1,d=*) (**) 5
(**) Consecutive primes in arithmetic progression (2,d=*) (**) 5
10545 Consecutive primes in arithmetic progression (3,d=*) 5 5
2575 Consecutive primes in arithmetic progression (4,d=*) 5 5
1072 Consecutive primes in arithmetic progression (5,d=*) 5 5
1000(*) Cullen primes 20 14
1070 Cunningham chain (16p+15) 5 5
(**) Cunningham chain (2p+1) (**) 5
1000(*) Cunningham chain (32p+31) (**) 1
10536 Cunningham chain (4p+3) 5 5
2355 Cunningham chain (8p+7) 5 5
(**) Cunningham chain (p) (**) 5
1029 Cunningham chain 2nd kind (16p-15) 5 5
47784 Cunningham chain 2nd kind (2p-1) 5 5
10014 Cunningham chain 2nd kind (4p-3) 5 5
1829 Cunningham chain 2nd kind (8p-7) 5 5
(**) Cunningham chain 2nd kind (p) (**) 5
143484 Divides Fermat 20 20
145123 Divides GF(*,10) 20 20
219096 Divides GF(*,12) 20 21
277717 Divides GF(*,3) 20 22
250522 Divides GF(*,5) 20 20
237790 Divides GF(*,6) 20 22
326724 Divides Phi 20 20
19632 ECPP 20 252
1343 Euler Irregular primes 20 20
1000(*) Factorial 20 19
5521 Fibonacci cofactor 20 20
1000(*) Fibonacci Number 20 11
4444 Fibonacci Primitive Part 20 20
1000(*) Gaussian Mersenne norm 20 15
338991 Generalized Cullen 20 20
1150678 Generalized Fermat 20 357
13028 Generalized Lucas Number 20 28
25140 Generalized Lucas primitive part 20 20
28856 Generalized Repunit 20 20
795940 Generalized Unique 20 75
359818 Generalized Woodall 20 20
1640 Irregular Primes 20 20
14897 Lehmer number 20 20
13319 Lehmer primitive part 20 20
5287 Lucas Aurifeuillian primitive part 20 20
5338 Lucas cofactor 20 20
1001 Lucas Number 20 20
11060 Lucas primitive part 20 20
39751 Mersenne 20 20
4365 Mersenne cofactor 20 20
360410 Near-repdigit 20 23
206365 Palindrome 20 20
9126 Partitions 20 20
1368 Primorial 20 20
3360 Quadruplet (1) 5 5
3360 Quadruplet (2) 5 5
3360 Quadruplet (3) 5 5
3360 Quadruplet (4) 5 5
1117 Quintuplet (1) 5 5
1117 Quintuplet (2) 5 5
1117 Quintuplet (3) 5 5
1117 Quintuplet (4) 5 5
1117 Quintuplet (5) 5 5
1000(*) Repunit 20 1
1037 Sextuplet 5 5
31112 Sophie Germain (2p+1) 20 20
31112 Sophie Germain (p) 20 20
10047 Triplet (1) 5 5
10047 Triplet (2) 5 5
10047 Triplet (3) 5 5
35851 Twin (p) 20 20
(**) Twin (p+2) (**) 20
4361 Unique 20 20
1000(*) Wagstaff 20 9
1000(*) Woodall Primes 20 18
  (*)   Less than the allowed number are known.
(**)  These primes do not make the list on their own merits, but make the list because a companion prime does (e.g., a 'Twin (p+2)' will be on the list if and only if the associate 'Twin (p)' prime is.
  ---   Database last updated: 2017-03-29 21:20:07.
Below are the comments that are currently tolerated in the official comment field, but which appear on the list only if the prime is already on the list for some other reason. Note that provers can add unofficial comments that appear on the individual prime's page, but not in the official comment field.
* old special cases (1), APR-CL assisted (1), Compositorial (1), Cyclotomy Proof (13), Mills' prime (1), Multifactorial (3)
The number in parenthesis is the number currently on the list.

(up)Why are there more than allowed of some forms?

What? Sometimes there are more primes on the list than the number allowed for that form? This happens for the following two reasons.

First, any prime in the top 5000 will automatically be archived, and sometimes there are many of the given form that fit there. When these primes get too small for the top 5000, they will be removed from the list. For example, we may not archive any of a certain form (such as generalized uniques), but there may be some on the list because they fit in the top 5000.

Second, a prime outside of the top 5000 may remain on the list due to another comment. For example, for a long time the only Mills' prime on the list was one of the largest known ECPP primes. It was the latter comment that allowed it to remain on the list.

Chris K. Caldwell © 1996-2017 (all rights reserved)