20121109, 04:46  #1 
"William"
May 2003
New Haven
23×103 Posts 
New Factor leaves a C168 Mersenne Composite
yoyo@home user Shawn Reimerdes found a P54 that reduced the unfactored primitive of 2^{1584}+1 from C222 to C168. This is the fourth known factor of this primitive. A P33 was previously found by ElevenSmooth. The remaining composite has had sufficient ECM to justify GNFS factoring. It is the smallest
Last fiddled with by wblipp on 20121109 at 23:57 Reason: Not Smallest Composite 
20121109, 07:07  #2 
Dec 2011
11×13 Posts 
Congrats to Shawn!!
Based on your posted progress data, a rough calculation suggests finding the next factor via ECM has an expected time of about 20000 coredays on an i7 2600K. Now that the number has been "cracked" down to 168 digits, a very rough estimate of GNFS difficulty is about 500 coredays on the same CPU. (I'm sure someone with more experience will correct my estimates.) And while the other thread acknowledges it isn't the smallest 2+ Table composite cofactor, it is clearly the smallest unfactored ElevenSmooth composite cofactor. I'm afraid it is much bigger than anything I've ever attempted, but I bet you will find a set of users/projects who have the capability of handling a 168digit GNFS and the interest in finding the last factor of a future Cunningham Table entry. I would be happy to contribute some cycles if there is a distributed effort. Last fiddled with by rcv on 20121109 at 07:21 
20121109, 07:38  #3 
Sep 2009
2·3·163 Posts 
Although here on MF, people usually switch to 15e above GNFS difficulty 162163, 14e can cope with difficulty 168. When the activity on RSALS burst by nearly an order of magnitude after I announced we were shutting down, I had the clients work on nearrepdigit 77771_259, GNFS difficulty 172. The polynomial used 31bit LPs, 2LP.

20121109, 13:02  #4  
"Åke Tilander"
Apr 2011
Sandviken, Sweden
2×283 Posts 
Quote:


20121114, 09:14  #5 
Sep 2009
2×3×163 Posts 
Shall NFS@Home's 14e factor this composite ?

20121114, 11:31  #6 
Dec 2011
11×13 Posts 
I vote "Yes", please. As already noted, this number is definitely ready for GNFS.
BTW, I added four other future 2table entries to the "sixthsmallest" thread referenced by wblipp. However, I don't know how much ECM has been performed on these numbers. 
20121121, 14:43  #7 
Sep 2009
3D2_{16} Posts 
In the next few days, I'll queue the composite cofactor of 2^1584+1 to NFS@Home's 14e, using the following polynomial provided by Greg Childers:
Code:
# norm 2.550152e16 alpha 6.342621 e 4.936e13 rroots 5 skew: 8702992.93 c0: 1114440592287328527195245530261195706821 c1: 420303723002969758765944488602445 c2: 1192641766822455254646269947 c3: 155159242787910119869 c4: 15269072427156 c5: 112056 Y0: 375237357010895436011270610824532 Y1: 4114903083139799 
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