# ECM candidates

This page lists some families of numbers which are potential targets for ECM. Some of those factorizations are useful for other applications, some are not. The main goal of this page is to give pointers to people maintaining lists of known factors, in order not to waste/duplicate the efforts.
• The Cunningham Project tries to factor numbers of the form an+1 or an-1 with a=2, 3, 5, 6, 7, 10, 11, 12, with bounded exponent n; Sam Wagstaff publishes lists of remaining composites and found factors, and Paul Zimmermann maintains a file with known factors and ecm effort. Will Edgington maintains a table of factors of Mersenne numbers. Paul Leyland maintains a similar table for exponent from 1200 to 10,000. Arjen Bot maintains a table of factors of 2n+1 for 1200 < n < 5000 and n prime.
• Fermat numbers are particular Cunningham numbers, and already got special effort. Wilfrid Keller gives known factors up to large indices.
• an+1 or an-1, 13 ≤ a ≤ 99. Richard Brent maintains tables of their factorizations, called Brent-Montgomery-te Riele (BMtR) tables. From January 1st, 2013, Jonathan Crombie now maintains those lists at http://myfactors.mooo.com/.
• factors of 2n + 2m + 1: see this web site maintained by Bill Gnadt. Sam Wagstaff estimated the number of primes of this form in this paper.
• homogeneous Cunningham numbers an+bn or an-bn: tables by Paul Leyland (no longer maintained) and in the factor database (search for an+bn for example)
• Koide Yousuke maintains a table of repunits factors (10n-1). Kamada Makoto maintains a table of near-repdigit numbers, which also contains factors of 10n+1 up to n=2000 (see also http://www.alfredreich.com up to n=5400).
• Fibonacci or Lucas numbers: Peter Montgomery and Ralf Stephan also work on their factorizations.
• Cullen and Woodall numbers, maintained by Wilfrid Keller and Paul Leyland;
• partition numbers, collected by Hisanori Mishima;
• Smarandache numbers, i.e. numbers of the form 1234567891011121314, or reverse ones of the form 1413121110987654321, collected by Ralf Stephan and Micha Fleuren, including symmetric and circular Smarandache numbers.
• Crandall numbers, of the form 2(q-1)/2+1 or 2(q-1)/2-1 (the one divisible by 3), especially for q=5807 (c793), 10501, 10691, 11279, 12391, 14479, 42737, 83339, and 95369;
• Wolstenholme numbers, i.e. numerators of 1+1/2+...+1/n, collected by Hisanori Mishima, who also maintains tables of (alternating) sums of factorials, Euler and Bernoulli numbers, products of primes minus the next prime; Sam Wagstaff also factored Euler and Bernoulli numbers;
• cyclotomic numbers: Hisanori Mishima is maintaining tables for φ(n) ≤ 100.