20180306, 14:16  #661 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×3^{3}×109 Posts 
C374 = P37 * C338 http://factordb.com/index.php?id=1100000000438534384

20180309, 17:28  #662 
Sep 2008
Kansas
2^{2}·7·11^{2} Posts 
These are the "easiest" ones in the t600 file. All are fully ECMed and ready for factMsieve or your favorite NFS suite. Be warned, they will take several weeks each on a four core system. All can use the degree halving polynomial.
Code:
SNFS216 C161 969550980294955841 12 SNFS221 C169 11658852700685942029849 10 SNFS218 C176 6721393100152677634549 10 SNFS222 C191 2959025653654433029 12 SNFS219 C193 1943777054011345723 12 SNFS219 C198 7560423642616328727781 10 SNFS211 C200 1372831999148013167419 10 SNFS214 C201 2642270235905971097617 10 SNFS224 C205 5056496574263951471 12 SNFS216 C212 4408320270589390433141 10 SNFS213 C214 563120390493837601 12 SNFS222 C219 16952580673293897376649 10 
20180309, 20:17  #663  
Jan 2018
3×11 Posts 
Quote:
Here are the last two full factorizations from my 25 digit work: C856 = P26 * PRP831 http://factordb.com/index.php?id=1100000000685532388 C848 = P27 * PRP821 http://factordb.com/index.php?id=1100000000511682066 

20180312, 02:36  #664 
Jan 2018
3·11 Posts 
Here are two more factorizations:
C157 = P40 * C118 The resulting composite is now the smallest composite in the t2100 file. C195 = P37 * P159 
20180313, 00:14  #665  
Oct 2007
Manchester, UK
2^{2}·3·113 Posts 
Quote:


20180314, 20:50  #666 
Dec 2017
3×23 Posts 
Sorry to break in but I've got a couple of questions. Firstly, is there a current uptodate list of the composites that need factoring? I've had a look at the t2100 file on Pascal's site and it's dated 07/2017. The first composite in that is only 103 digits, assuming I've understood the format correctly. That would be only hours' work for GNFS, but there's no entry in factordb for it. What am I missing?
And secondly, is there an explanation for the source of some of these numbers? It's all very well being asked to factor a^2+a+1 for some large a, but in some circumstances it might be possible to use the algebraic form of a to generate a better polynomial for SNFS. Thanks for any help Steve 
20180314, 21:47  #667  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
16065_{8} Posts 
As far as I know, pascal's lists + cross referencing the FDB (+ maybe checking this thread) is the best way to get up to date information.
As for algebraic form, all OPN numbers are of the form sigma(p^(m)) ~ (p^(m+1) + 1)/(p1) for prime p and various m. Quoting the description: Quote:
Code:
32255735115003306513692416251654907813239666644230888161178016067166921037 2 520265584590898068859754378133743032119120450966287262997866771153575366825960463761001173652178932891073 Last fiddled with by Dubslow on 20180314 at 21:55 

20180315, 16:20  #668  
Jan 2018
33_{10} Posts 
Quote:
Any composites in the t2100 file that are less than 130 digits are usually factored very quickly and they will be removed when the file is updated. However, primes found when factoring other composites can become a new unfactored branch in the proof tree and will be added to the file. The p for the C103 that you mentioned is the second largest factor of (93120567780622842301313419400053^71) and it looks like that factorization was finished on February 16th. That was a longwinded of saying that the t2100 file is a fairly current list of the numbers that need factoring but it's easy to think that it's out of date if you just look at the smallest numbers in the file because every time a branch in the proof tree is fully factored it can lead to new small composites being added to the t2100 file. 

20180315, 21:07  #669 
Dec 2017
3×23 Posts 
Thank you both. That's a lot clearer now.
It would still be useful to know the source of the base, I think. For instance, if we want to factor N=a^2+a+1, that's not very useful as a polynomial for SNFS. However, if we know that a=b^4+b^3+b^2+b+1, we can substitute that to get a nice sextic with root b. Probably teaching my grandmother to suck eggs. Cheers Steve 
20180315, 22:14  #670 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Often the FDB can be used (in a somewhat roundabout way) to find where a particular base came from. For instance, search for the base of our standard example, the C103 which is p^2+p+1=sigma(p^2), where p is 322555...; click on the "More information" arrow, where it tells you that p is a factor of some (ex)C143; clicking "More information" there reveals that this C143, and thus p, is also a factor of a (ex)C177; notice that the special form of this C177 is given in the search box; you can either use that form directly, or click "More information" and follow to the C185, which the filledin search box tells us is "(4664607259009421338832033924593^71)/4664607259009421338832033924592". So we see that our p in question is in fact the largest factor of this number, which is sigma(4664607259009421338832033924593^6). You can of course repeat the process on this previousbase 4664607259009421338832033924593, to see where it came from, etc, etc as far back as you please. It's a bit tedious, but doable.
Last fiddled with by Dubslow on 20180315 at 22:14 
20180316, 15:04  #671 
Jan 2018
33_{10} Posts 
I'm now running 35 digit ECM on the ~3,500 C1XX composites in the t2100 file. I've decided to stop posting all of the full factorizations that I find and to just post factorizations that lead to a composite with less than 130 digits. I don't have a factoring program installed but I'm hoping that someone else will be able to quickly finish the factorization.
C143 = P38 * C106 http://factordb.com/index.php?id=1100000000824425381 
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