Extensions to Cunningham tables

[Raman]

The extensions to Cunningham 3- tables are available at this website.
http://homes.cerias.purdue.edu/~ssw/cun/xtend/index.html
3+ tables as well as other tables will be too extended soon.

People can try out with ECM upon these numbers, or also GNFS/SNFS upon those easy candidates.

How are the factors being used up?

> From raman22feb1988@gmail.com   Thu Jun  3 10:20:26 2010
> Subject: How are factors used?
> To: Samuel S Wagstaff <ssw@cerias.purdue.edu>
>
> May I know how people make use of that Cunningham factors in other
> mathematical works? Where are they used?

See Section 2 of my paper
http://homes.cerias.purdue.edu/~ssw/cun1.pdf
for uses of Cunningham factors.
You can search MathSciNet for papers that refer to the Cunningham
Book (Brillhart et al.) to see more uses of Cunningham factors.

> You told that for extending the tables, you wait for customer  requests.
> Who are those customers?

Some people need the prime factors of 3^n +- 1 (and perhaps other bases,
too) to construct elliptic curves with nice properties for cryptography.

> How do you drive traffic into your website, how do you make people
> visit your site often? If I build up with my own website, then how
> can I drive people into my own site, especially that right people?

That is a trade secret; I will never tell.

> After that 6,355- 6,365- has been done
> I hope that 6,349- would be in your next "Most wanted" lists :)
> as well as 3,563+ after that 3,562+ has been over, the next hole is
> only being at exponent 589 for that base 3 tables actually
> What about that for 3,563- 3,569- truncating up with that base 3  tables...

No Most Wanted number has been done yet on Page 117.  The new Most  Wanted
list for the next page is likely to be the same as the old one.

> Time to extend up with that base 3 tables...
> For current limits of that Cunningham tables, base 6,7,8,10 are up  to
> index 400 while that for base 9 is only actually upto index 300

The Cunningham tables are being extended and you can help.  See
http://homes.cerias.purdue.edu/~ssw/cun/xtend/index.html

> At least those multiples of 3 would be easier beyond 600 index for  that
> base 3 tables If you respect that Aurifeuillian extension tables  for
> those numbers splitting up into two parts that are easily being
> factorable, then why not respect those splitting up into 3 parts  that
> are easily factorable, as being two thirds part of that original  number
> itself?  Such as for example
> 3,603- 3,621- 3,627- 3,633- 3,651- 3,657- 3,675- 3,681- 3,687-  3,699-
> 3,636+ 3,654+ 3,678+ 3,696+

Most of these (all the -) are listed on the page
http://homes.cerias.purdue.edu/~ssw/cun/xtend/index.html
The others (+) will be added soon.

> Sometimes, right now, even multiples of 5, 7, 11, 13 for base 3  tables
> extension beyond index 600 are being easily accessible in order to  be
> factorable, like those of 3,611- I would say that rather this base 3
> tables are indeed strongly lagging behind those of that other  tables,
> really only actually, right now, by now itself, thus

I agree.  The 3+ table will be extended soon.

> After that, even extension for that base 2 tables will provide  provision
> in order for others to do work upon that Mersenne Numbers with no  known
> prime factors list of numbers, that is candidates like
> 2,1237- 2,1277-
> after that enclosed list of numbers such as those of
> 2,1123+
> for which no non-primitive prime factors is being known up so right  now,
> at all, first of all as well as 2,1061- with no known prime factors  at
> all, thus indeed really

All of the Cunningham tables will be extended, even the base 2 tables.

Keep the factors coming!

Sam Wagstaff

Edit: Yes, of course that I requested for permission in order to share this up from Prof. Sam Wagstaff

[R.D. Silverman]

Originally Posted by Raman

The extensions to Cunningham 3- tables are available at this website.
http://homes.cerias.purdue.edu/~ssw/...end/index.html
3+ tables as well as other tables will be too extended soon.

People can try out with ECM upon these numbers, or also GNFS/SNFS upon those easy candidates.

I hope not. What fun is there in doing numbers that are easy?
Let's make more progress on the current tables before extending them.

As soon as the tables are extended, people will go after the low hanging fruit
and ignore the current tables.

[XYYXF]

On the other hand, people with smaller CPU power would help on the extended tables, while on the current tables they won't give any help.

[R. Gerbicz]

And it looks like that sam doesn't use/know the factordb.com, because using that: p=10739917689782934291348634826555020368414408001; is a factor of 3^775-1 found/added by Jonathan Crombie on March 4, 2010, 12:25 am.

(by hand I've checked that there is no more factor on this extension table, and added some to the factor database.)

[Batalov]

I wonder if anyone (Paul?) knows how much these targets are ECMed.
Probably well above 40 digits.

I'll do 3,705- for a warmup in one quad-day (diff.180). There's also 3,777- (diff.211), a nice sextic.

[bdodson]

Originally Posted by R.D. Silverman

I hope not. What fun is there in doing numbers that are easy?
Let's make more progress on the current tables before extending them.

As soon as the tables are extended, people will go after the low hanging fruit
and ignore the current tables.

Repeated inquiries about the length of the smaller of the 18 tables didn't
get hardly any interest in extending the tables. I wonder whether having
the Smaller-but-Needed numbers headed into the c180's triggered the
extension this time? I can't recall ever having seen the "2nd smallest
available" (& etc, 1st, 3rd ...) listed on the "who's" before, which suggests
that Sam was giving some of his attention to that.

We used to have similar bumps in the available easy numbers back in
the RSA partition Challenge --- factoring the last number in a 10-digit
range would bring a bunch of new numbers up into the range where
points were given (0 points --> 1 points, 1 points ---> 2). Depending
upon how many new numbers are in the pending extension (base-2, even!),
I wonder whether much of the lower low-hanging fruit will have cleared
in a few months, and we can see where things settle. -Bruce

PS -- Maybe we also needed some new Cunningham 2^n-1's for PS3
search targets?

[chris2be8]

3^607-1 is one of the odd perfect roadblocks and has had a lot of ECM run against it, see http://www.mersenneforum.org/showthr...991#post202991 for details. Also 3^661-1, 3^857-1, 3^971-1 and 3^991-1. So don't start running curves shorter than B1=260M against 3^607-1or shorter B1=43M against any of the rest.

It would be nice if the tables said how much ECM etc has been run against unfactored numbers to prevent wasted effort.

Chris K

[R.D. Silverman]

Originally Posted by bdodson

Repeated inquiries about the length of the smaller of the 18 tables didn't
get hardly any interest in extending the tables. I wonder whether having
the Smaller-but-Needed numbers headed into the c180's triggered the
extension this time? I can't recall ever having seen the "2nd smallest
available" (& etc, 1st, 3rd ...) listed on the "who's" before, which suggests
that Sam was giving some of his attention to that.

We used to have similar bumps in the available easy numbers back in
the RSA partition Challenge --- factoring the last number in a 10-digit
range would bring a bunch of new numbers up into the range where
points were given (0 points --> 1 points, 1 points ---> 2). Depending
upon how many new numbers are in the pending extension (base-2, even!),
I wonder whether much of the lower low-hanging fruit will have cleared
in a few months, and we can see where things settle. -Bruce

PS -- Maybe we also needed some new Cunningham 2^n-1's for PS3
search targets?

Why can't they go after the remaining 2+ and 2LM's???

[Raman]

Originally Posted by R.D. Silverman

I hope not. What fun is there in doing numbers that are easy?
Let's make more progress on the current tables before extending them.

As soon as the tables are extended, people will go after the low hanging fruit
and ignore the current tables.

I will be posting up with the extended tables over Mersenne wiki along with current tables in beautiful format shortly, along with sorted difficulty levels separately.

I agree that the extended tables will be a target for people with limited resources as well as a practice tool for newcomers. I think that it is time to expand your world of tables, rather than focusing upon a small room of Cunningham candidates. 3,607- 3,661- have no known non-trivial factors at all.

Let us make use of this thread for extended Cunningham table reservations as well as posting up with new factors? What about that of the extended 3LM tables?

Please explain how factoring up with

help in order to find out new odd perfect numbers (if any does really exist up, in any case), or clear roadblocks or possibilities upto a certain limit.

[R.D. Silverman]

Originally Posted by Raman

I will be posting up with the extended tables over Mersenne wiki along with current tables in beautiful format shortly, along with sorted difficulty levels separately.

I agree that the extended tables will be a target for people with limited resources as well as a practice tool for newcomers. I think that it is time to expand your world of tables, rather than focusing upon a small room of Cunningham candidates.

There are over 400 remaining.

Please explain how factoring up with

help in order to find out new odd perfect numbers .

It won't. But it can help raise the lower bound.

[xilman]

Originally Posted by R.D. Silverman

Why can't they go after the remaining 2+ and 2LM's???

("They" are presumably are Arjen Lenstra's group at EPFL.) There are several possible answers to this question. The first is that computations modulo 2- are very easy on a binary computer. It's clear that their current software has been written to exploit this.

Another answer is that they haven't (yet) re-coded their arithmetic library to exploit the more difficult but still simple reduction modulo 2+.

A third answer may be that they don't want to. There are still plenty of 2- numbers which have not yet been tested to the p75 level.

You should be able to think of other plausible answers.

Paul