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Old 2010-02-02, 21:04   #1
fivemack
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Default What do I have to do to break free?

9336 iteration 1037 splits as 1119687698...<144> = 2^6 · 7 · 127^2 · 12577 · 11466743 · 86572333 · 1241124045...<118>

but nonetheless iteration 1038 has the 127^1 still there, still mocking me with its evil set of seven set bits. What is the probability of driver escape at this level?

(similarly, I've had 31^2 at iterations 3916, 3930 and 3962 of 6822, but 31^1 returns the very next line)

Last fiddled with by fivemack on 2010-02-02 at 21:06
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Old 2010-02-03, 07:24   #2
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Quote:
Originally Posted by fivemack View Post
9336 iteration 1037 splits as 1119687698...<144> = 2^6 · 7 · 127^2 · 12577 · 11466743 · 86572333 · 1241124045...<118>

but nonetheless iteration 1038 has the 127^1 still there, still mocking me with its evil set of seven set bits. What is the probability of driver escape at this level?

(similarly, I've had 31^2 at iterations 3916, 3930 and 3962 of 6822, but 31^1 returns the very next line)
The 2s count of a prime number is the exponent of the highest power of 2 that divides that prime + 1. To escape 2^6 * 127, firstly, the 127 must be raised to an even power. Also, the sum of the 2s counts of all the other prime factors must be no greater than 6.
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Old 2010-02-04, 15:45   #3
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The 7 term is the evil one. It gives 3 factors of 2 and will stick around as long as some term is 6 mod 7.

I would like to see how many factors of two the 'large terms' (ones that don't stick around) add for the different number of digits. It might be possible to calculate that - or at least a good estimate. The average is higher than 3 however which means that with the 7 term you are unlikely to escape.

Last fiddled with by Greebley on 2010-02-04 at 15:46
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Old 2010-02-05, 16:45   #4
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Quote:
Originally Posted by Greebley View Post
The 7 term is the evil one. It gives 3 factors of 2 and will stick around as long as some term is 6 mod 7.

I would like to see how many factors of two the 'large terms' (ones that don't stick around) add for the different number of digits. It might be possible to calculate that - or at least a good estimate. The average is higher than 3 however which means that with the 7 term you are unlikely to escape.
Well, val_2(p+1) has mean 2, the mean number of prime factors for a number around N is (theorem of Erdos) loglog N, and I think is expected to be Poisson-distributed with mean loglog N; so around 10^100 you'd expect there to be about 5.5 prime factors; 2 and 127 account for two of those, leaving about 3.5 big ones each contributing about two factors of two. So the numbers get big, and escape is unlikely.
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Old 2010-02-05, 18:15   #5
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Quote:
Originally Posted by fivemack View Post
Well, val_2(p+1) has mean 2, the mean number of prime factors for a number around N is (theorem of Erdos) loglog N, and I think is expected to be Poisson-distributed with mean loglog N; so around 10^100 you'd expect there to be about 5.5 prime factors; 2 and 127 account for two of those, leaving about 3.5 big ones each contributing about two factors of two. So the numbers get big, and escape is unlikely.
I would expect more like 5 ≈ log(log(1e100 / 254)) - 1/2 - 1/127 other prime factors, if they behave like random multiples of 254 near 1e100.
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Old 2010-03-07, 07:41   #6
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From 20916:
Code:
 670 .   c117 = 2^6 * 3 * 11 * 13 * 127^2 * 84584688662069 * 303457268457331 * 23305983260885734320137249 * 3127792565339118661368476021978704179191553219477805861
 671 .   c118 = 2^6 * 3 * 11 * 83 * 127 * 839 * c107
And not only did it gain a digit that line, that cursed 3 has been turning up the heat (16 lines so far)!

* Line stolen from Bill Clinton [inside joke for the USAians here]
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Old 2010-04-08, 13:08   #7
Mini-Geek
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How do I lose 2^5 * 7 (not a listed driver or guide, but it seems like a guide to me)? It's taken over sequence 652296 for 75 lines (going up 11 digits) so far.

Last fiddled with by Mini-Geek on 2010-04-08 at 13:08
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Old 2010-04-08, 18:11   #8
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Quote:
Originally Posted by Mini-Geek View Post
How do I lose 2^5 * 7 (not a listed driver or guide, but it seems like a guide to me)? It's taken over sequence 652296 for 75 lines (going up 11 digits) so far.
I lost it here: http://factordb.com/search.php?se=1&...r=1446&to=1448
At line 1447, the 2^5 switched to 2^3, but it still had 7^2 as a factor. Line 1448 lost the 7 as well, so now the only guide is 2^3, so this big sequence is now in a downward slope.
Still leaves the question of how 2^5 * 7 (or 7 to any other power) can be broken directly? Or can it?

Last fiddled with by Mini-Geek on 2010-04-08 at 18:11
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Old 2010-04-08, 18:47   #9
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It is a listed guide: http://www.lafn.org/~ax810/analysis.htm
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