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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Re: New Factorization Record!!!
Message-ID: <485@faron.UUCP>
Date: Sun, 23-Feb-86 15:57:29 EST
Article-I.D.: faron.485
Posted: Sun Feb 23 15:57:29 1986
Date-Received: Wed, 26-Feb-86 04:06:04 EST
References: <479@faron.UUCP> <1189@oddjob.UUCP>
Distribution: net
Organization: The MITRE Coporation, Bedford, MA
Lines: 51

> In article <479@faron.UUCP> bs@faron.UUCP (Robert D. Silverman) writes:
> >A new factoring record has just been set: I have factored a 75 digit cofactor
> >    128
> >of 6   + 1 using an enhanced version of Carl Pomerance's Quadratic Sieve
> >Algorithm. [ ...factorisation follows ...]
> >For the ambitious among you I present a list of the numbers currently
> >on the '10 MOST WANTED' list to be factored: (Cxx means a composite number 
> >of xx digits)
> >
> >     512
> >(1) 2   + 1 = 2424833.C148
> >
> >     227   114
> >(2) 2   - 2   + 1 = 5.C68
> >
> [......]
> >      149
> >(10) 3   + 1 = 4.C71
> >
> >Bob Silverman
>      This is fascinating. Is it possible to explain where these composite
> numbers come from? Were/are they bound to have few large factors, or has that
> been established by trial and error? Are they all extremely simple base n
> (for some small n) for number-theoretic, computational or sociological
> reasons? (i.e. are they known to be likely to have few large factors, or are
> they much easier to work with, or are they just the sort of numbers people are
> likely to look at?) And is there a committee somewhere which decides the 
> 'most wanted' list? :-)
>                          Adrian Kent

They come from the 'Cunningham Project'. This is a project started at the
turn of the century by Allen Cunningham, an English military officer with
an interest in number theory. The project is attempting to factor b^n + 1
and b^n - 1 for bases b = 2, 3, 5, 6, 7, 10, 11, and 12, and various n
which depend upon the base b. The American Mathematical Society has published
a book on the project entitled 'Factorizations of b^n +/- 1 for b = 2, 3...12
up to high powers'. It is volume 22 in the 'Contemporary Mathematics'
series. The most wanted list is kept by mathematicians on the project.
Usually John Selfridge, who is editor of Math Reviews decides which numbers
go on the list when one is knocked off.

By the way, I have already finished (2) on the list, Someone in Germany
did (10) and I am currently working on (3).

Various investigators have attacked these numbers using a variety of methods.
It is a virtual certainty that none have any factors under 20 digits.

Anyone who is seeking more info should feel free to write me.

Bob Silverman
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