Top Twenty's Home Page

(Definitions of Archivable Form and Class)

The Prime Pages maintains the database of the largest known primes. This collection started when Samuel Yates listed all of the primes with at least 1,000 digits and called them titanic primes [Yates84]. By 1996 that list had grown to 50,000 primes, and was growing by roughly 1,000 primes each month. Unfortunately, most of these new primes were just barely larger than the original 1,000 digits--people were seeking more primes, but not larger primes!

At that point we decided to return the focus to record primes by just keeping the 5000 largest known primes (and every prime that has ever made this list). We also included up to twenty each of certain selected forms--even if they do not make the top 5000. We call these forms archivable. Which primes are these? In 1997, after a great deal of debate, we settled on the following definition.

An archivable form of prime is one which is the subject of more than one mathematical journal article written by more than one set of authors. Two articles from a single author are sufficient if at least one is in a major refereed journal such as Math. Comp., otherwise four articles from single authors are required. To prove a number is archivable, one need only supply the article references. The Prime Pages will keep the top 5000 primes plus up to twenty (20) of each of the archivable primes.

The fact that we do not keep a certain prime (smaller than the 5000th) does not make it unworthy or uninteresting; it just means we do not keep records of that form of prime.

In September 2000, it became necessary to make a ruling on certain classes of forms of primes such as the primes in arithmetic progression and Cunningham chains. We adopted the following rule.

Sometimes the form itself is not archivable, but the primes of that form belong to a larger class that is. We call these archivable classes. For example, primes in arithmetic progressions are an archivable class, so we will keep up to five each of the terms in an arithmetic progressions of primes starting with the third term. Other examples include: Cunningham chains, Cunningham chains of the second kind, triplets, quadruplets and quintuplets. We keep the top five (not the top twenty) of selected terms in an archivable class.

We will eventually make a Top 20 page for each of the archivable forms. If we are missing your favorite form, then just e-mail us the appropriate references (or write the articles if necessary). The Top 5000 list only stores primes with at least 1,000 digits, so the forms are restricted to those for which we know large examples. This excludes excellent forms such as the Wilson primes and Wieferich primes. All known primes of these two types are already listed in the Prime Glossary.

Finally, there are some classes of primes we list for historical reasons (such as they were so commented on Yates' lists). These are tolerated on the current list, and only appear on the list if the prime there for some other reason.

Use the sizes tab above to see how large a prime needs to be to make the list.