# How big is big enough?

## 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 515131 digits. This is increasing at roughly 50,000 digits per year. Click on the trends tab above to view the change over the last few 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 thousands of digits more!

## 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
425548 Arithmetic progression (3,d=*) 5 10
25992 Arithmetic progression (4,d=*) 5 10
10377 Arithmetic progression (5,d=*) 5 10
3019 Arithmetic progression (6,d=*) 5 6
2271 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
10546 Consecutive primes in arithmetic progression (3,d=*) 5 5
3021 Consecutive primes in arithmetic progression (4,d=*) 5 5
1209 Consecutive primes in arithmetic progression (5,d=*) 5 5
1000(*) Consecutive primes in arithmetic progression (6,d=*) 5 1
1000(*) Cullen primes 20 14
1070 Cunningham chain (16p+15) 5 5
(**) Cunningham chain (2p+1) (**) 5
1000(*) Cunningham chain (32p+31) (**) 1
10713 Cunningham chain (4p+3) 5 5
2972 Cunningham chain (8p+7) 5 5
(**) Cunningham chain (p) (**) 5
1141 Cunningham chain 2nd kind (16p-15) 5 5
63634 Cunningham chain 2nd kind (2p-1) 5 5
10014 Cunningham chain 2nd kind (4p-3) 5 5
2272 Cunningham chain 2nd kind (8p-7) 5 5
(**) Cunningham chain 2nd kind (p) (**) 5
176118 Divides Fermat 20 20
275977 Divides GF(*,10) 20 20
340432 Divides GF(*,12) 20 20
451509 Divides GF(*,3) 20 23
371306 Divides GF(*,5) 20 20
404389 Divides GF(*,6) 20 20
455479 Divides Phi 20 20
27630 ECPP 20 269
2051 Euler Irregular primes 20 20
2188 Factorial 20 20
7213 Fibonacci cofactor 20 20
1000(*) Fibonacci Number 20 13
6082 Fibonacci Primitive Part 20 20
1000(*) Gaussian Mersenne norm 20 16
396957 Generalized Cullen 20 20
2545761 Generalized Fermat 20 1642
16185 Generalized Lucas Number 20 28
25140 Generalized Lucas primitive part 20 20
42480 Generalized Repunit 20 21
1067588 Generalized Unique 20 82
417693 Generalized Woodall 20 20
3734 Irregular Primes 20 20
16625 Lehmer number 20 20
15537 Lehmer primitive part 20 20
8202 Lucas Aurifeuillian primitive part 20 20
6941 Lucas cofactor 20 20
1001 Lucas Number 20 20
12324 Lucas primitive part 20 20
227832 Mersenne 20 20
8649 Mersenne cofactor 20 20
544907 Near-repdigit 20 23
220285 Palindrome 20 21
11138 Partitions 20 20
1750 Primorial 20 20
1543 Quintuplet (1) 5 5
1543 Quintuplet (2) 5 5
1543 Quintuplet (3) 5 5
1543 Quintuplet (4) 5 5
1543 Quintuplet (5) 5 5
1000(*) Repunit 20 2
1002 Septuplet 5 5
1037 Sextuplet 5 5
43080 Sophie Germain (2p+1) 20 20
43079 Sophie Germain (p) 20 20
11050 Triplet (1) 5 5
11050 Triplet (2) 5 5
11050 Triplet (3) 5 5
46873 Twin (p) 20 20
(**) Twin (p+2) (**) 20
7168 Unique 20 20
1000(*) Wagstaff 20 10
1000(*) Woodall Primes 20 19
(*)   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: 2022-07-05 11:50:05
* old special cases (1), APR-CL assisted (1), Cyclotomy Proof (15), Multifactorial (4)

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), Cyclotomy Proof (15), Multifactorial (4)

The number in parenthesis is the number currently on the list.

## 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.