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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).

(*) 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.