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Old 2013-03-05, 08:33   #1
arbooker
 
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"Andrew Booker"
Mar 2013

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Default second Euclid-Mullin sequence

I am pleased to report that the 14th term of the second Euclid-Mullin 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 GMP-ECM, 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 co-factor. Bearing in mind that the 13th term of the sequence was computed by Wagstaff 20 years ago, I won't hold me breath!
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Old 2013-03-05, 09:04   #2
xilman
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Quote:
Originally Posted by arbooker View Post
I am pleased to report that the 14th term of the second Euclid-Mullin sequence (A000946 in the OEIS) is the following P101:
26402590817665123115124196783110486814361930234455788059710183484151247460960172672371287819122033451
Nice work, especially finding the large ECM factor.
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Old 2013-03-05, 13:47   #3
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Excellent! The c229 was in one of my low-priority ECM rotations, but I'm embarassed to say how little progress I had made. I suggest you send the c62 to Zimmerman for possible inclusion in this page.
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Old 2013-03-05, 15:13   #4
fivemack
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Good work!
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Old 2013-03-05, 15:32   #5
R.D. Silverman
 
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Quote:
Originally Posted by arbooker View Post
I am pleased to report that the 14th term of the second Euclid-Mullin 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 GMP-ECM, 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 co-factor. Bearing in mind that the 13th term of the sequence was computed by Wagstaff 20 years ago, I won't hold me breath!
Nicely done.

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?
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Old 2013-03-05, 15:49   #6
arbooker
 
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Quote:
Originally Posted by R.D. Silverman View Post
This sequence isn't as interesting as the first sequence, however.
Why? Because it has been shown to omit infinitely many primes.
Yes, I was aware of that. To my dismay, the new term did not lead to a certificate that 79 is omitted (except for 2, 3, 7 and 43, all primes <= 73 are omitted, as follows from knowledge of the first 13 terms and the method of Cox and Van der Poorten).

Quote:
Originally Posted by R.D. Silverman View Post
Query: Is the following question even decidable?

Given a prime p, is it a member of this sequence?
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).
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Old 2013-03-05, 20:20   #7
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Quote:
Originally Posted by arbooker View Post
I ran a little ecm on the next composite to crack
How much?

Quote:
Bearing in mind that the 13th term of the sequence was computed by Wagstaff 20 years ago, I won't hold me breath!
You never know. I thought the same about this C392, which caught my eye a while ago. Turned out to be P43*P349. Yours might break the same way.
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Old 2013-03-05, 20:34   #8
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Congratulations. I also had been burning a few cycles on this number.
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Old 2013-03-05, 21:28   #9
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I've updated my page for EM-seqs with the latest found here.
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Old 2013-03-06, 01:44   #10
arbooker
 
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Quote:
Originally Posted by Mr. P-1 View Post
How much?
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:
You never know. I thought the same about this C392, which caught my eye a while ago. Turned out to be P43*P349. Yours might break the same way.
That would be nice.
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Old 2013-03-06, 03:40   #11
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Is there a standard B1 for 70-digit 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 every-other level, or half the expected number of curves at each level, but never skipping directly to t65-searching.

-Curtis
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