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 545311 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 form or class alone).

digits required	archivable form or class	number archived	number on list
(**)	Arithmetic progression (1,d=*)	(**)	10
(**)	Arithmetic progression (2,d=*)	(**)	11
807954	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
1197	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
70021	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
340147	Divides GF(*,10)	20	20
364696	Divides GF(*,12)	20	20
551637	Divides GF(*,3)	20	21
427044	Divides GF(*,5)	20	20
535087	Divides GF(*,6)	20	21
455479	Divides Phi	20	20
32218	ECPP	20	274
2069	Euler Irregular primes	20	20
2188	Factorial	20	20
7213	Fibonacci cofactor	20	20
1000(*)	Fibonacci Number	20	13
6822	Fibonacci Primitive Part	20	20
1000(*)	Gaussian Mersenne norm	20	16
396957	Generalized Cullen	20	20
3289511	Generalized Fermat	20	1447
17264	Generalized Lucas Number	20	28
25140	Generalized Lucas primitive part	20	20
42480	Generalized Repunit	20	21
1067588	Generalized Unique	20	78
417693	Generalized Woodall	20	20
3821	Irregular Primes	20	20
16625	Lehmer number	20	20
15537	Lehmer primitive part	20	20
8202	Lucas Aurifeuillian primitive part	20	20
7317	Lucas cofactor	20	20
1001	Lucas Number	20	20
12324	Lucas primitive part	20	20
227832	Mersenne	20	20
9736	Mersenne cofactor	20	20
608990	Near-repdigit	20	23
220285	Palindrome	20	21
13210	Partitions	20	20
1750	Primorial	20	20
3503	Quadruplet (1)	5	5
3503	Quadruplet (2)	5	5
3503	Quadruplet (3)	5	5
3503	Quadruplet (4)	5	5
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
43091	Sophie Germain (2p+1)	20	20
43091	Sophie Germain (p)	20	20
11050	Triplet (1)	5	5
11050	Triplet (2)	5	5
11050	Triplet (3)	5	5
47722	Twin (p)	20	20
(**)	Twin (p+2)	(**)	20
7200	Unique	20	20
1000(*)	Wagstaff	20	12
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: 2023-03-31 11:37:08

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 (2)

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.