Homogeneous Cunningham Numbers

"Homogeneous Cunningham numbers" is the term for numbers of the form an ± bn which seem not otherwise to have a name in the literature. Cunningham numbers proper take the form an ± 1, where a is an integer, not a prime power, between 2 and 12 inclusive. The tables below contain factorizations for b < a12 and gcd (a, b) = 1. The tables for 9n ± 4n are omitted because they are but subsets of the tables for 3n ± 2n

These tables were originally created by Bob Silverman. In April 2006, Bob sent them to Paul Leyland, who made them public and began to maintain them. With the factoring work of many people around the world, very few of the original numbers remained unfactored by the start of 2007. At that point some of the tables were extended to higher index, most of which were completed by mid-2009. A further extension was largely completed by March 2016, whereupon Paul and Jon Becker made an extension that set the upper limit on an ± bn to 1024 bits. Many relatively small factors were found by Jon and Paul before the extension tables were published. In 2018 Jon added Aurifeuillian factorizations to the tables, with the individual Aurifeuillian constituents also going up to a limit of 1024 bits. In October 2018 Jon took over maintenance of the tables.

The tables below contain Bob's original data, together with all extensions and the factors supplied by numerous workers, split into separate files and converted to a format designed to be both human-readable and easily parsed by programs.

Additions and corrections to the tables are welcome, and a number of people have already contributed further factorizations. Please send all such data to Jon. The file UPDATE contains a list of changes to the files made since 1 April 2016 and here are those reported before that date.

Tom Womack runs a reservation system for these numbers, so that workers who prefer NFS to ECM do not inadvertently repeat each other's work. Please note: Tom's page does not forward any factorizations reported to it, so please remember to send your results to Jon by email.

An ECMnet server (for version 2 clients only) is running on ecm.unshlump.com:8194. A status page for the server can be found here. It is updated once per day.

\    
a  \  b
    \
2 3 4 5 6 7 8 9 10 11
3 3n-2n
3n+2n
                 
4   4n-3n
4n+3n
               
5 5n-2n
5n+2n
5n-3n
5n+3n
5n-4n
5n+4n
             
6       6n-5n
6n+5n
           
7 7n-2n
7n+2n
7n-3n
7n+3n
7n-4n
7n+4n
7n-5n
7n+5n
7n-6n
7n+6n
         
8   8n-3n
8n+3n
  8n-5n
8n+5n
  8n-7n
8n+7n
       
9 9n-2n
9n+2n
    9n-5n
9n+5n
  9n-7n
9n+7n
9n-8n
9n+8n
     
10   10n-3n
10n+3n
      10n-7n
10n+7n
  10n-9n
10n+9n
   
11 11n-2n
11n+2n
11n-3n
11n+3n
11n-4n
11n+4n
11n-5n
11n+5n
11n-6n
11n+6n
11n-7n
11n+7n
11n-8n
11n+8n
11n-9n
11n+9n
11n-10n
11n+10n
 
12       12n-5n
12n+5n
  12n-7n
12n+7n
      12n-11n
12n+11n

ECM users may find a gzipped list of all the composite cofactors useful. There are 900 numbers in this file.