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How big is big enough?
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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 140293 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!
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 |
| 100354 |
Arithmetic progression (3,d=*) |
5 |
10 |
| 10043 |
Arithmetic progression (4,d=*) |
5 |
10 |
| 5132 |
Arithmetic progression (5,d=*) |
5 |
6 |
| 1303 |
Arithmetic progression (6,d=*) |
5 |
5 |
| 1287 |
Arithmetic progression (7,d=*) |
5 |
5 |
| 1000(*) |
Arithmetic progression (8,d=*) |
5 |
2 |
| (**) |
Consecutive primes in arithmetic progression (1,d=*) |
(**) |
5 |
| (**) |
Consecutive primes in arithmetic progression (2,d=*) |
(**) |
5 |
| 5132 |
Consecutive primes in arithmetic progression (3,d=*) |
5 |
5 |
| 1050 |
Consecutive primes in arithmetic progression (4,d=*) |
5 |
5 |
| 1000(*) |
Consecutive primes in arithmetic progression (5,d=*) |
5 |
1 |
| 1000(*) |
Cullen primes |
20 |
14 |
| (**) |
Cunningham chain (2p+1) |
(**) |
5 |
| 4543 |
Cunningham chain (4p+3) |
5 |
5 |
| 1208 |
Cunningham chain (8p+7) |
5 |
5 |
| (**) |
Cunningham chain (p) |
(**) |
5 |
| 17105 |
Cunningham chain 2nd kind (2p-1) |
5 |
5 |
| 4564 |
Cunningham chain 2nd kind (4p-3) |
5 |
5 |
| 1057 |
Cunningham chain 2nd kind (8p-7) |
5 |
5 |
| (**) |
Cunningham chain 2nd kind (p) |
(**) |
5 |
| 64217 |
Divides Fermat |
20 |
20 |
| 82754 |
Divides GF(*,10) |
20 |
20 |
| 109204 |
Divides GF(*,12) |
20 |
22 |
| 167371 |
Divides GF(*,3) |
20 |
25 |
| 144451 |
Divides GF(*,5) |
20 |
23 |
| 91697 |
Divides GF(*,6) |
20 |
23 |
| 109163 |
Divides Phi |
20 |
20 |
| 8046 |
ECPP |
20 |
198 |
| 1000(*) |
Euler Irregular primes |
20 |
11 |
| 1000(*) |
Factorial |
20 |
13 |
| 1701 |
Fibonacci cofactor |
20 |
20 |
| 1000(*) |
Fibonacci Number |
20 |
10 |
| 2943 |
Fibonacci Primitive Part |
20 |
20 |
| 1000(*) |
Gaussian Mersenne norm |
20 |
12 |
| 125569 |
Generalized Cullen |
20 |
20 |
| 405516 |
Generalized Fermat |
20 |
130 |
| 12865 |
Generalized Lucas Number |
20 |
27 |
| 22754 |
Generalized Lucas primitive part |
20 |
20 |
| 14534 |
Generalized Repunit |
20 |
20 |
| 205824 |
Generalized Unique |
20 |
166 |
| 100274 |
Generalized Woodall |
20 |
20 |
| 1000(*) |
Irregular Primes |
20 |
16 |
| 11967 |
Lehmer number |
20 |
20 |
| 9013 |
Lehmer primitive part |
20 |
20 |
| 2407 |
Lucas Aurifeuillian primitive part |
20 |
20 |
| 1292 |
Lucas cofactor |
20 |
20 |
| 1000(*) |
Lucas Number |
20 |
15 |
| 3853 |
Lucas primitive part |
20 |
20 |
| 25962 |
Mersenne |
20 |
20 |
| 1813 |
Mersenne cofactor |
20 |
20 |
| 69847 |
Near-repdigit |
20 |
20 |
| 105015 |
Palindrome |
20 |
22 |
| 4366 |
Partitions |
20 |
20 |
| 1007 |
Primorial |
20 |
20 |
| 1157 |
Quadruplet (1) |
5 |
5 |
| 1157 |
Quadruplet (2) |
5 |
5 |
| 1157 |
Quadruplet (3) |
5 |
5 |
| 1157 |
Quadruplet (4) |
5 |
5 |
| 1000(*) |
Quintuplet (1) |
5 |
3 |
| 1000(*) |
Quintuplet (2) |
5 |
3 |
| 1000(*) |
Quintuplet (3) |
5 |
3 |
| 1000(*) |
Quintuplet (4) |
5 |
3 |
| 1000(*) |
Quintuplet (5) |
5 |
3 |
| 1000(*) |
Repunit |
20 |
1 |
| 20013 |
Sophie Germain (2p+1) |
20 |
20 |
| 20013 |
Sophie Germain (p) |
20 |
20 |
| 5132 |
Triplet (1) |
5 |
5 |
| 5132 |
Triplet (2) |
5 |
5 |
| 5132 |
Triplet (3) |
5 |
5 |
| 24342 |
Twin (p) |
20 |
20 |
| (**) |
Twin (p+2) |
(**) |
20 |
| 2136 |
Unique |
20 |
20 |
| 1000(*) |
Wagstaff |
20 |
8 |
| 1000(*) |
Woodall Primes |
20 |
18 |
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(*) 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: 2009-11-07 19:20:25.
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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 (5), Cyclotomy Proof (23), Mills' prime (1), Multifactorial (3), Tetradic (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.