20050523, 21:16  #1 
Feb 2004
France
1623_{8} Posts 
A question about Cunningham tables
Hi,
I've looked at Cunningham tables and I found no factorisation of big numbers like: 2^29800963  1 = 15436497912585268249 * ... or: 2^4782969 + 1 = 3635056441 * ... Why ? Is the goal to fill the tables from the beginning up only ? Why not also collecting the factors of big numbers ? I've not found a clear description of the exact goals of this project. Tony 
20050523, 21:29  #2  
Nov 2003
1D24_{16} Posts 
Quote:
tables are beyond our ability at the upper end. Extending them to such large exponents would be pointless, because factoring numbers that size are totally a matter of (a GREAT DEAL) of luck. We only succeed when there are 1 or 2 small factors, and the cofactor is prime. Such numbers are extremely rare. So a systemic effort would simply yield a sprinkling of small factors where there is no hope of finishing off the cofactors. May I suggest that you read the introduction to the tables (available on Sam Wagstaff's Website) to understand their history???? The current goal is to finish the existing tables. Sometimes when a subtable gets finished, or close to being finished, it is extended by a modest amount. What is "big" depends on best available algorithms. By current or nearfuture standards, "big" is any general number of more than about 200 digits, or a Cunningham number of more than 250 digits. Currently *hopeless* is a general number of 768 bits or a Cunningham number of more than about 1024 bits (SNFS could just barely do a 1024 bit number with a massive, massive effort) 

20050524, 04:20  #3  
"William"
May 2003
New Haven
3·787 Posts 
Quote:
http://www.garlic.com/~wedgingt/mersenne.html 

20050524, 16:27  #4 
Aug 2002
Buenos Aires, Argentina
544_{16} Posts 
It would be nice to have an appendix of the Cunningham project that lists the known _complete_ factorization of a^b +/ 1 with bases 2 to 12 and exponents greater than the ones used actually in the project.
That list would not very extensive, because the pseudoprime cofactors should be verified. That should limit the values of a^b +/ 1 to less than, say, 5000 digits at most, unless someone discovers a method to find quickly if a factor of a^b +/ 1 is prime or not. 
20050524, 16:43  #5 
Feb 2004
France
3·5·61 Posts 
Thanks for the answers.
 I've warned Will about the Mersenne factor.  I've read again the introduction to the Cunningham tables but it still is unclear for me. Probably due to my poor english.  Thanks for the explanation of the goal of the project. It's clear now. Nevertheless, I think it is so fun to find a factor of so big numbers, even by luck (or with prime95 for the first one and with PARI/gp for the second one)). Where is collected all the knowledge about the factorization properties of these Cunningham numbers ? Regards, Tony 
20050524, 17:23  #6  
"Mark"
Apr 2003
Between here and the
3^{3}×233 Posts 
Quote:


20050524, 17:50  #7  
Nov 2003
2^{2}×5×373 Posts 
Quote:
It isn't a bad idea. Why don't you put one together? 

20050524, 18:02  #8  
Nov 2003
16444_{8} Posts 
Quote:
If you find it fun, you are certainly welcome to do it. However, I share the view of the Cunningham authors: "there is a real feeling of accomplishment in breaking apart some huge number which has withstood assaults for decades, especially when in doing so one has had to devise and carry out some new computational scheme" I never saw the fascination in just mindlessly running code written by others. I wanted to understand the algorithms and write the code myself. Therein lies the real reward: writing and using your OWN code. As for finding small factors of very large numbers, I think it is just too easy. The real reward comes from the success arising from an *extended* effort. And let me repeat in parting what we are all told by our parents: Finish what you start before going on to something new....... 

20050524, 18:23  #9  
Nov 2003
2^{2}×5×373 Posts 
Quote:
as John Selfridge put it: "Picking plums at waist height". It is a first step toward complete factorizations, but the CURRENT tables already go beyond the stateoftheart. They are more than adequate for pushing developments in algorithms. I see no reason why finding a small factor of 2^n1 (for large n) through luck, using ECM, contributes to improving the art. Note: I would have more respect for your endeavors if you were using your own code.. While I too use GMPECM, I have in the past coded my own ECM program. I devote my time to improving my NFS code, rather than improving my ECM code. A single person can't optimize everything. 

20050524, 19:33  #10  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2^{3}×11^{3} Posts 
Quote:
I certainly don't have time to do it myself, but I may be able to assist in a small way. Paul 

20050524, 21:23  #11 
Aug 2002
Buenos Aires, Argentina
2^{2}×337 Posts 
Thanks, Bob and Paul.
I actually don't have time either but for base 2 someone can convert the information located at the complete factorization page of Will Edgington Mersenne site to the format used in the Cunningham project. Many records with general primality testing algorithms were done with large factors of nonprime Mersenne numbers. 
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