"Homogeneous Cunningham numbers" is the term I use for those 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 12 ≥ a > b 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 supplied to me by Bob Silverman in April 2006. Very few of the original numbers remained unfactored by the start of 2007 and so some of the tables were extended to higher index; most of which were completed by mid-2009. A further extension was largely completed by 2016-03-16 whereupon the upper limit on an ± bn was raised to 1024 bits. Many relatively small factors were found by Jon Becker and myself before the extension tables were published. In 2018 we added Aurifeuillian factorizations to the tables, with the individual Aurifeuillian constituents also going up to a limit of 1024 bits.
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 my standard format, designed to be both human-readable and easily parsed by programs.
I welcome additions and corrections to the tables 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 to me 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 18.104.22.168:8195.
a \ b
ECM users may find a gzipped list of all the composite cofactors useful. There are 1450 numbers in this file.
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