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
a^{n}+1 or a^{n}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 2^{n}+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.
 a^{n}+1 or a^{n}1, 13 ≤ a ≤ 99.
Richard Brent maintains
tables of their factorizations, called BrentMontgomeryte Riele
(BMtR) tables.
From January 1st, 2013, Jonathan Crombie now maintains those lists at
http://myfactors.mooo.com/.
 factors of 2^{n} + 2^{m} + 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 a^{n}+b^{n} or a^{n}b^{n}:
tables by Paul Leyland (no longer maintained) and in the factor database (search for a^{n}+b^{n for example)
}
 Koide Yousuke maintains a table
of repunits factors (10^{n}1).
Kamada Makoto maintains a table of
nearrepdigit numbers, which also contains factors of
10^{n}+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^{(q1)/2}+1 or
2^{(q1)/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.
 primorial
numbers, collected by
Lorenzo Allegrucci;
 factorials plus
or minus one, collected by Andrew Walker;

k*2^{n}+/1, collected by Mikael Klassons;
 home primes
sequences,
i.e. repeated factorizations of concatenated prime factors;

x^{y}+y^{x}, collected by Andrey Kulsha;
 the
ElevenSmooth project aims at finding large factors by ECM or P1.

ECM is also used in primality proving using methods of Konyagin and
Pomerance, or BrillhartLehmerSelfridge tests:
Lehmer and
Lucas primes.