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-   -   no twin left behind? (https://www.mersenneforum.org/showthread.php?t=13382)

 gd_barnes 2010-05-07 19:33

[quote=science_man_88;214272]I'm using mousekeys because my mouse stopped working so outside of speech it's the only way I know to have a way to click thinks.[/quote]

Just buy a new mouse. Good gawd. They cost < \$5.

 gd_barnes 2010-05-07 19:54

[quote=Mini-Geek;214280]Status update: I'm currently at n=119k. I've found a decent number of twins that aren't in Karsten's list, but the largest is only 42932385 832 ([URL="http://factordb.com/search.php?id=165336191"]42932385*2^832-1[/URL] and [URL="http://factordb.com/search.php?id=165336190"]+1[/URL] for FactorDB entries), which at 259 digits is not even close to making any notable twin prime lists. I'll send the full list of total and new twins when I'm done checking all numbers.[/quote]

Nice work Tim. This is an interesting effort. I did something similar a little over 2 years ago, but not for ALL of the primes on Karsten's pages. I think I got up to something like k=10000. Of course there are many more primes on there now so I'm sure you'll find plenty of new ones that Karsten hasn't been able to check yet.

The main thing that I found is that they just "die off" for n>~1500.

BTW, one hint that you may have already realized: You only need to check k's divisible by 3. It's not possible for a -1 and +1 prime for k's that are not divisible by 3. Either k*b^n-1 or k*b^n+1 will have a factor of 3.

Another interesting effort would be to check k*b^n-1 primes for k*b^n-3 primes to make a twin in that manner. Of course you could not prove anything that would make the top 20 but you probably could prove n<=10K in a reasonable amount of time and the chances are very small that you would find anything larger than that anyway.

Gary

 science_man_88 2010-05-07 20:45

way to ruin my fun GD I was just going to post about that though I related 3 slightly differently (no surprise on that). I got it through (p+1/2^n)*2^n-1, must switch to (p-1/2^n)*2^n+1. I found that p gave a x in my comparison that was always a multiple of 3*2^n all minus 1. Anyways for now PM me if you want more info if I don't figure anything new out.

 Mini-Geek 2010-05-07 23:46

Status: n=283k. 471 numbers remaining to test, ignoring any future sieving (I've got it sieved to 600M and am ready to break off n=300k-400k). Nothing new to report. I've only confirmed primes.
[quote=gd_barnes;214296]The main thing that I found is that they just "die off" for n>~1500.[/quote]
Yeah, they thin out a LOT after about there. I've found 3229 twin primes below n=1500 and 62 above it (counting the one at n=333333, which I haven't rediscovered yet; also this is not NEW twins, this is ALL twins from the file that I know about).
[quote=gd_barnes;214296]BTW, one hint that you may have already realized: You only need to check k's divisible by 3. It's not possible for a -1 and +1 prime for k's that are not divisible by 3. Either k*b^n-1 or k*b^n+1 will have a factor of 3.[/quote]
I knew this, but ignored it because I knew srsieve would find the factor of 3 for such numbers (and thus pick up on it, if only dumbly), even if it took a little longer to figure it than needed.
[quote=gd_barnes;214296]Another interesting effort would be to check k*b^n-1 primes for k*b^n-3 primes to make a twin in that manner. Of course you could not prove anything that would make the top 20 but you probably could prove n<=10K in a reasonable amount of time and the chances are very small that you would find anything larger than that anyway.[/quote]
This could be done too, but I think it'd be harder to sieve, test, and verify. But it does have the plus that there have probably not been as many people searching it. Someone (maybe me) might want to do it after I do the +1 check. But one thing at a time. :smile:

 Mini-Geek 2010-05-08 12:01

1 Attachment(s)
Status: complete to n=400k. No surprises so far.
Previously I've been running on one core only, now I'm running n=400k+ on all 4 cores, with manual LLR with the work split roughly evenly (figured it'd be less trouble than getting PRPnet to run it...and I don't really care if the slowest core, or largest work chunk, takes 50% longer than the shortest one, because I've got PRPnet running below it to use my CPU when it can, and with work like this such a thing would mean hours longer to wait, not days).
The candidates become very thin after about n=700000 (something like how the twins become very thin after about n=1500). This may be due to the search depths of efforts to find Riesel primes, the sparseness of primes that large, or a combination (I'd guess most likely a combination based mostly on the search depths). e.g. there are only 27 candidates for n>700k remaining after sieving to 2G. For comparison, there are 254 with 400k<n<700k, and 662 in 100k<n<400k)
The odds of a twin in that few candidates is exceedingly slim. But there's always a chance. And it's much less effort than directly looking for a twin that large. Unfortunately, it means you're limited to a small search space, until people hurry up and find more huge primes! :razz:
Even though I don't expect to find any large primes now, it's not as if the work was all wasted. Now everyone can know that the Riesel primes as of 2010-05-07 were checked for twins, and that they don't need to duplicate that work. Plus it's helped fill in twin primes in Karsten's list. :smile: This check should be redone now and then with the new primes. To that end, the current list of primes should be archived somewhere. Karsten?

Would anyone like me to post all of the results, for possible doublechecking?

Some (interesting?) trivia: the median n (median=half are smaller, half are larger) for all known primes of the form k*2^n-1 (assuming Karsten's list is exhaustive, which is a close enough assumption) is 923. The median n for all known twin primes of the form k*2^n±1 (assuming my current list of twins is exhaustive, which is also a close enough assumption) is 15.

Since the odds of finding another twin are so slim, I'll post the lists now.
This is the list of new twin primes: (ones that I found that were not on Karsten's list)
[code]1164735 1
1587045 1
5154945 1
14421825 1
18544365 1
26277285 1
65737005 1
94708605 1
1770403635 1
20887630335 1
67740637965 1
202635 2
1082115 2
1105845 2
2447445 2
2448855 2
17179515 2
26277285 2
31400085 2
42932385 2
1355509155 2
2794023375 2
52873540005 2
323169 3
589305 3
1669995 3
2448855 3
31400085 3
47915595 3
94708605 3
99311355 3
1355509155 3
43455956115 3
3710369067405 3
237915 4
422427 4
1303035 4
1669995 4
2222415 4
2332125 4
5128905 4
13867245 4
25308855 4
39598845 4
60627945 4
74784765 4
76298145 4
2995125705 4
666945 5
979125 5
1303035 5
2448855 5
3536385 5
18544365 5
49619895 5
2305225065 5
3234846615 5
4042850955 5
52890983145 5
101413071045 5
979125 6
1164735 6
2448855 6
8311875 6
18544365 6
31400085 6
1810322085 6
1885495755 6
2794023375 6
43455956115 6
1164735 7
5128905 7
5775915 7
14421825 7
39418665 7
56591535 7
2995125705 7
944007 8
42932385 8
5128905 9
5169615 9
56591535 9
1810322085 9
1587045 10
2995125705 10
3234846615 10
52890983145 10
20049 11
17179515 12
88335225 13
76220123265 13
101413071045 13
26277285 14
76220123265 14
1164735 15
6553365 15
1064887395 15
69234960795 15
237915 16
3271575 16
2995125705 16
43455956115 16
52873540005 17
59463215955 17
13935645 18
1105845 19
74784765 19
3234846615 19
72992835135 20
1164735 21
4849845 21
944007 22
39598845 22
323169 25
5169615 27
72992835135 27
14361 29
420147 30
99311355 30
1946388015 31
5128905 33
3710369067405 33
323169 37
8311875 37
67740637965 38
323169 39
20049 41
6021525 41
1946388015 43
1105845 44
25308855 45
2305225065 45
1105845 46
104598726375 51
46650333015 52
3234846615 57
48797387925 61
76298145 64
3710369067405 69
74784765 79
2995125705 84
104598726375 92
420147 110
6021525 121
13867245 127
4849845 132
2889081195 133
3710369067405 150
237123 154
5154945 162
52654604145 201
59463215955 207
4849845 272
8311875 295
8311875 411
52873540005 443
69234960795 459
59463215955 461
99311355 464
56591535 468
2305225065 485
67740637965 668
104598726375 761
42932385 832
[/code]The list of all twin primes I found is attached (note the next paragraph, though:).

srfile changed "58644190679703485491635 29" to "1991269380820904371 29". I think srfile must have truncated it to 16 hexdigits (64 bits), as in hex those k's are c6b1ba268f39ff129b3 and 1ba268f39ff129b3, respectively.
58644190679703485491635*2^29±1 are indeed prime. 1991269380820904371*2^29±1 are not prime. Because of this bug, 58644190679703485491635 29 is not on my list of twins, but is really a twin. (I only discovered this while writing this post)

 science_man_88 2010-05-08 13:55

good work mini by the way I got my mouse working again switched ports which let windows download a driver.

 Mini-Geek 2010-05-08 17:01

Status: various, but ~over half done (n=454k is the status of 400k-500k)
[quote=Mini-Geek;214361]the median n for all known primes of the form k*2^n-1 is 923.[/quote]
Correction: 919. I forgot to sort the list, and Karsten's list has some very large k's prime's out-of-order. This pushed the apparent median to the incorrect value of 923.

 petrw1 2010-05-08 17:12

 kar_bon 2010-05-08 22:06

I've just updated the Summary pages at [url=www.rieselprime.de]RieselPrimeDatabase[/url] with the last 164 missing Twin primes so far.

Tim, if you find some more, please post here, too.

 Mini-Geek 2010-05-09 11:58

I've completed the search with no more primes found.
Again, if anybody wants results or anything, they're available on request. Factors aren't, as I used srsieve and didn't tell it to write the factors out (it just removed them from the sieve file).

 Mini-Geek 2010-11-17 22:02

I reran this on the list of primes as of 2010-10-06 (only running on the difference between this and the earlier list, which was 2010-05-07) and found just one new twin:
1071495*2^49±1 are prime (21 digits)

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