20130305, 08:33  #1 
"Andrew Booker"
Mar 2013
3×29 Posts 
second EuclidMullin sequence
I am pleased to report that the 14th term of the second EuclidMullin sequence (A000946 in the OEIS) is the following P101:
26402590817665123115124196783110486814361930234455788059710183484151247460960172672371287819122033451 It is notable that this is smaller than the 13th term, making it the second known decrease in the sequence. (The first, between the 9th and 10th terms, was found by Naur in 1984.) For the record, the other prime factors of 1+(product of the first 13 terms) are as follows: 11, 13, 107536547, 78476577792946809375725792668447, 14078867962762048764039308139541671900484125027527542153799653, 9348432970765876153321791268740642151733897733787681100617533565963 The P62 factor was found by GMPECM, and the final two P67 and P101 were split by yafu/msieve/ggnfs. Thanks to the authors of these programs for making such factorizations possible (and fun!). I ran a little ecm on the next composite to crack and found the prime factors 155400913, 279619159, 55573207945331425309351, with a C332 cofactor. Bearing in mind that the 13th term of the sequence was computed by Wagstaff 20 years ago, I won't hold me breath! 
20130305, 09:04  #2 
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2^{2}×11^{2}×23 Posts 
Nice work, especially finding the large ECM factor.

20130305, 15:13  #4 
(loop (#_fork))
Feb 2006
Cambridge, England
2^{2}×1,613 Posts 
Good work!

20130305, 15:32  #5  
Nov 2003
1D24_{16} Posts 
Quote:
This sequence isn't as interesting as the first sequence, however. Why? Because it has been shown to omit infinitely many primes. Query: Is the following question even decidable? Given a prime p, is it a member of this sequence? 

20130305, 15:49  #6  
"Andrew Booker"
Mar 2013
3·29 Posts 
Quote:
Cox and Van der Poorten conjectured that their method will always work to determine this. The conjecture is almost certainly true, but may itself be undecidable. One can at least prove that if there is no such procedure then the sequence must be very thin (in the precise sense that it has logarithmic density zero in the primes). 

20130305, 20:20  #7  
Jun 2003
7×167 Posts 
How much?
Quote:


20130305, 20:34  #8 
Aug 2004
New Zealand
11100001_{2} Posts 
Congratulations. I also had been burning a few cycles on this number.

20130306, 01:44  #10  
"Andrew Booker"
Mar 2013
3×29 Posts 
I ran 192 curves at 85e7 with no luck. I'll put some more effort into it, perhaps as far as a full t65, and give up after that.
Quote:


20130306, 03:40  #11 
"Curtis"
Feb 2005
Riverside, CA
2·2,579 Posts 
Is there a standard B1 for 70digit factor searches? I did some experiments with EM47 a few months ago, and found 29e8 more efficient than 25e8 when using 6GB memory, but lost interest before I did a more thorough set of experiments.
arbooker Did you search any smaller B1 bounds? I've heard of people doing everyother level, or half the expected number of curves at each level, but never skipping directly to t65searching. Curtis 
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