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

Mini-Geek 2010-05-07 11:58

no twin left behind?
 
I was just thinking about this again:
[URL]http://www.mersenneforum.org/showthread.php?t=12591[/URL] (looking for twin primes by searching the 'other side', i.e. swapping -1 and +1, of a known prime)
But the top 5000 list has so many different prime forms, and is only accessible through text or HTML (both of which would not be terribly easy to extract the essentials from) it'd be non-trivial to get a list of candidates to test. So then I thought of NPLB: tons of primes, all in the form k*2^n-1, all in a central DB that can pick out the k and n for me. Easy!
I'd like to apply this idea to [URL="http://www.noprimeleftbehind.net/stats/index.php?content=prime_list"]the full list of NPLB primes[/URL]. NPLB has found 6604 (as of this writing) primes for -1, and I'd like to find if any of those have a twin on the +1 side. I could try to download all the pages and extract the k and n, but I figured it'd be easier to ask one of you with DB access to dump the list for me. :smile: (consider it requested :smile: preferably in the NewPGen format of one "k n" per line)
6604 candidates of all sizes before any sieving is sure to reduce to a handful of testable candidates. TBH I'm not too hopeful that such a twin does exist, but it'd be such an easy check it's almost a waste to look for new primes to find their twins before checking old primes to see if they have twins.

kar_bon 2010-05-07 13:24

You can also obtain an even greater list from my RieselPrimeDatabase for download [url=http://www.rieselprime.de/Downloads.htm]here[/url]!

I've just updated this list (and the twins-list, too). It's in the npg-format sorted by n-values for all primes in all summary-pages.

science_man_88 2010-05-07 13:55

stupid mouse failure lol (using mousekeys).

(10:46) gp > for(n=1,10,for(k=1,10,if(isprime(k*2^n-1) && isprime(k*2^n+1),print(k"*"2"^"n"+ or -1"))))
2*2^1+ or -1
3*2^1+ or -1
6*2^1+ or -1
9*2^1+ or -1
1*2^2+ or -1
3*2^2+ or -1
9*2^3+ or -1
6*2^5+ or -1
3*2^6+ or -1
9*2^7+ or -1
(10:46) gp >

this what you want ?

kar_bon 2010-05-07 14:54

[QUOTE=science_man_88;214256]this what you want ?[/QUOTE]

No, I think not!

Those small twins are all known (see list at download or the summary-pages).

He was asking of higher n-levels say n~500000 for any prime of the form k*2^n-1.
Test those k/n-pairs if the number k*2^n+1 is prime, too! The Riesel-side (-1) is known to be prime for many values here but none has ever tested the '+1'-side!
OTOH the chance for finding a new twin by testing only these primes is very low.

mdettweiler 2010-05-07 15:12

[quote=Mini-Geek;214240]I was just thinking about this again:
[URL]http://www.mersenneforum.org/showthread.php?t=12591[/URL] (looking for twin primes by searching the 'other side', i.e. swapping -1 and +1, of a known prime)
But the top 5000 list has so many different prime forms, and is only accessible through text or HTML (both of which would not be terribly easy to extract the essentials from) it'd be non-trivial to get a list of candidates to test. So then I thought of NPLB: tons of primes, all in the form k*2^n-1, all in a central DB that can pick out the k and n for me. Easy!
I'd like to apply this idea to [URL="http://www.noprimeleftbehind.net/stats/index.php?content=prime_list"]the full list of NPLB primes[/URL]. NPLB has found 6604 (as of this writing) primes for -1, and I'd like to find if any of those have a twin on the +1 side. I could try to download all the pages and extract the k and n, but I figured it'd be easier to ask one of you with DB access to dump the list for me. :smile: (consider it requested :smile: preferably in the NewPGen format of one "k n" per line)
6604 candidates of all sizes before any sieving is sure to reduce to a handful of testable candidates. TBH I'm not too hopeful that such a twin does exist, but it'd be such an easy check it's almost a waste to look for new primes to find their twins before checking old primes to see if they have twins.[/quote]
If you want I can dump the DB's list for you rather easily, but it sounds like Karsten's list might be even better, since it contains ALL known Riesel primes rather than just NPLB's. If you'd still like to get the DB's list, though, let me know.

science_man_88 2010-05-07 15:34

[QUOTE=kar_bon;214260]No, I think not!

Those small twins are all known (see list at download or the summary-pages).

He was asking of higher n-levels say n~500000 for any prime of the form k*2^n-1.
Test those k/n-pairs if the number k*2^n+1 is prime, too! The Riesel-side (-1) is known to be prime for many values here but none has ever tested the '+1'-side!
OTOH the chance for finding a new twin by testing only these primes is very low.[/QUOTE]

I am annoyed by mousekeys and it may take hours for my pari code to do this if I could use message boxes in win32 assembly good I'd try that.

the hard part is a prime check loop I think if I use Iczelion's win32 ASM tutorials. I did buy a ASM book though but it's for HLA might have to adapt it to win32, or someone else can try this as I suck at it lol.

mdettweiler 2010-05-07 15:47

[quote=science_man_88;214268]I am annoyed by mousekeys and it may take hours for my pari code to do this if I could use message boxes in win32 assembly good I'd try that.[/quote]
If you don't mind my asking, why bother using MouseKeys in the first place (since if it's that annoying, I'm assuming you don't need it for accessibility reasons)? :huh:

science_man_88 2010-05-07 15:50

[QUOTE=mdettweiler;214271]If you don't mind my asking, why bother using MouseKeys in the first place (since if it's that annoying, I'm assuming you don't need it for accessibility reasons)? :huh:[/QUOTE] 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.

Mini-Geek 2010-05-07 15:53

[quote=kar_bon;214253]You can also obtain an even greater list from my RieselPrimeDatabase for download [URL="http://www.rieselprime.de/Downloads.htm"]here[/URL]!

I've just updated this list (and the twins-list, too). It's in the npg-format sorted by n-values for all primes in all summary-pages.[/quote]
That should work great, thanks. :smile: I'll see if I can find any twins that aren't already known.
[quote=science_man_88;214256]this what you want ?[/quote]
No, what Karsten said.
[quote=mdettweiler;214261]If you want I can dump the DB's list for you rather easily, but it sounds like Karsten's list might be even better, since it contains ALL known Riesel primes rather than just NPLB's. If you'd still like to get the DB's list, though, let me know.[/quote]
Unless for some reason there're any primes in the DB but not in Karsten's list, no I don't need NPLB's list. Thanks anyway.

Mini-Geek 2010-05-07 17:55

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.

kar_bon 2010-05-07 18:00

[QUOTE=Mini-Geek;214280]I've found a decent number of twins that aren't in your list, but the largest is only 42932385 832 (...) I'll send the full list of total and new twins when I'm done checking all numbers.[/QUOTE]

Good job! So I can eliminate some missings/errors from my database then!

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

[url]http://archives.tcm.ie/businesspost/2008/05/18/story32830.asp[/url]

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.

I've not yet updated the Download list of all Twin primes.

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)

Mini-Geek 2010-11-26 16:17

I just finished checking on the list of Proth primes for Riesel counterpart twins for the 2010-11-24 list. No new primes found.

Mini-Geek 2010-11-29 16:44

16294579238595022365*2^7±1 are prime (22 digits)
I've finished checking the Riesel list of 2010-11-26. This was the only new twin prime.

kar_bon 2010-11-29 19:54

[QUOTE=Mini-Geek;239184]16294579238595022365*2^7±1 are prime (22 digits)
I've finished checking the Riesel list of 2010-11-26. This was the only new twin prime.[/QUOTE]

This twin was in the 'old' list, too, as I can see.
I've nothing changed on the [url=http://www.rieselprime.de/Data/10e10.htm]10^10 Summary page[/url] since 2010-10-18.

Mini-Geek 2010-11-29 20:00

[QUOTE=kar_bon;239227]This twin was in the 'old' list, too, as I can see.
I've nothing changed on the [url=http://www.rieselprime.de/Data/10e10.htm]10^10 Summary page[/url] since 2010-10-18.[/QUOTE]

Oh, yeah you're right. Not sure how that happened...perhaps I accidentally included some numbers for testing that shouldn't have been, so when I found that I thought it was new and didn't properly check if it existed.

kar_bon 2010-11-29 20:05

[QUOTE=Mini-Geek;239230]Oh, yeah you're right. Not sure how that happened...perhaps I accidentally included some numbers for testing that shouldn't have been, so when I found that I thought it was new and didn't properly check if it existed.[/QUOTE]

I've checked my saved history-files and this twin is included since 2008-10-01.

But thanks for the checking of the twin list.

Mini-Geek 2011-01-17 16:10

1 Attachment(s)
[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]

I am checking for -3 (from Riesel primes) and +3 (from Proth primes) twins for the Riesel list of 2010-11-26 and the Proth list of 2010-11-24. I have found 4470 primes or PRPs so far, the largest being 5789*2^15513+3 (4674 digit PRP, and the Primo primality test is queued; of course 5789*2^15513[B]+1[/B] is a known prime, so this makes a twin PRP/prime pair). None so far are of a reportable size, either for being a top 10000 PRP or being a top 20 twin. I'm sure most of the small ones are already known, just because they're so small you wouldn't have to notice their somewhat special form to run across them, but there are still a good number that are hundreds to thousands of digits that were most likely not previously known. All primes are attached, in two files. The ones in pfgw-prime.log were trivially proven to be prime by PFGW, the ones in pfgw.log were found to be PRP. Of the 1773 PRPs there, I have verified the 1755 smallest and submitted the certificates to the FactorDB. (it actually took close to 30 of those and verified the rest itself; it doesn't bother taking outside certificates for primes under 300 digits, but it does enter it into its system for it to verify) The rest are still running in Primo.
The PRP testing is currently at n=475K. There are 275 candidates remaining. In the event (with a dismally low chance of under 3%) that one of those is prime, I'll have found a probable twin prime. AFAIK that would only be officially recognized as a non-twin PRP on the top PRPs list, as it would be too large to prove, but it would still be highly likely to be the largest pair of twin primes known.

henryzz 2011-01-17 19:55

What is the record for a non +-1 twin prime? Is there a type non +-1 that is easy to prove and sieve?

Mini-Geek 2011-01-17 21:50

[QUOTE=henryzz;247062]What is the record for a non +-1 twin prime? Is there a type non +-1 that is easy to prove and sieve?[/QUOTE]

The [URL="http://primes.utm.edu/top20/page.php?id=1"]top 20 twin primes[/URL] list consists solely of k*b^n+-1 (all but one have b=2). I don't know of any non +-1 that are easy to prove, because AFAIK efficient primality proving algorithms are generally based on N+1 or N-1 being factored to 33.33% (or PFGW's 'combined' test that uses largely-but-not-quite-33% factored N+1 and N-1, and of course things like Primo/ECPP that can work on any, but I wouldn't call ECPP 'efficient' in this context), so for twin primes, choosing (smooth number)+-1 is a clear choice. I don't know if a record list exists for non +-1 twin prime, or (probably the same) a pair of twin primes/PRPs, where at least one is only known to be a PRP. Given the obscurity of that, I'd guess my 5789*2^15513+1, +3 pair have a good chance of being that.
Status:
PRPing: n=513365, 209 candidates remain, sieving more before continuing.
proving PRPs: n=4414, 11 candidates remain. (over 1 hour per test; I'm not sure I'll go all the way with this, 2^15513 is starting to look pretty large...might take a few days just for that test)

henryzz 2011-01-18 17:09

If you give in before 5789*2^15513+1, +3 then I would do it. This one is worth testing because it is the largest we know of in its class.

Mini-Geek 2011-01-18 18:18

[QUOTE=henryzz;247207]If you give in before 5789*2^15513+1, +3 then I would do it. This one is worth testing because it is the largest we know of in its class.[/QUOTE]

Ok, I'll leave that one to you. I am also not proving 7027*2^13017-3 and 755*2^13474-3. You or anyone else can do those, or they could stay just PRPs for a while (until someone wants to prove them).
For clarity, here are the largest PRPs I found and who has them reserved for proving:
[CODE][URL="http://factordb.com/index.php?id=1100000000291772821"]517*2^6098+3[/URL] and below Mini-Geek (done, certificates in DB)
[URL="http://factordb.com/index.php?id=1100000000291801599"]7315*2^6423-3[/URL] Mini-Geek (done, certificate in DB)
[URL="http://factordb.com/index.php?id=1100000000291801677"]1381*2^6512+3[/URL] Mini-Geek (done, certificate in DB)
[URL="http://factordb.com/index.php?id=1100000000291801683"]7027*2^13017-3[/URL] [unreserved]
[URL="http://factordb.com/index.php?id=1100000000291801687"]755*2^13474-3[/URL] [unreserved]
[URL="http://factordb.com/index.php?id=1100000000287449188"]5789*2^15513+3[/URL] henryzz
[/CODE]The first 1767 (all before 517*2^6098+3) took about 19 hours. The last one I completed, 517*2^6098+3, took a little over 5 hours. I'd guess the next two I'm doing are 6-8 hours each and the largest three are several days to a few weeks each (unless you do multicore Primo or a distributed ECPP).
Also, a status on my PRPing:
I have split it into 3 parts and am running it on Prime95, (multiple cores and automatic P-1 makes it much better than PFGW when the numbers are this size :smile:) the lowest is at n=819630, and there are 29 candidates left to PRP. The largest one alone will probably take a couple days, since it is so large: 2^13466917-3. If that turns out PRP, that would be incredible! Besides being the largest PRP, with 4053946 digits, it would (probably) be a twin with a Mersenne prime. Unfortunately that is extremely unlikely...

Mini-Geek 2011-01-20 11:53

1 Attachment(s)
[CODE][URL="http://factordb.com/index.php?id=1100000000291801677"]1381*2^6512+3[/URL] and below Mini-Geek (done, certificates in DB)
[URL="http://factordb.com/index.php?id=1100000000291801683"]7027*2^13017-3[/URL] [unreserved]
[URL="http://factordb.com/index.php?id=1100000000291801687"]755*2^13474-3[/URL] [unreserved]
[URL="http://factordb.com/index.php?id=1100000000287449188"]5789*2^15513+3[/URL] henryzz
[/CODE]
Done PRPing all candidates and proving all candidates I said I would. All results I saved (started a little after n=20K) along with all primes in pfgw.log or pfgw-prime.log according to current proven status are attached.

2^13466917-3 would have taken a little over two days, but I put it on two cores for most of it, so it took closer to one day. Because it wasn't sieved very well, (only to 5 billion, or about 2^32) Prime95 chose P-1 bounds that gave it a 20% chance of finding a factor. Unfortunately it did not find a factor, even with such generous bounds, so I had to test it. Alas, the largest known twin Mersenne prime (i.e. 2^p-1 and (2^p-3 or 2^p+1) are prime) is just p=5: 29 and 31.
Just for fun, here are all known primes that are twin Mersenne or Fermat primes:
[CODE]2^16+1, +3 (65537, 65539)
2^4+1, +3 (17, 19)
2^5-1, -3 (29, 31)
3, 5, and 7, by various formulas (3=2^1+1=2^2-1, 5=2^1+3=2^2+1=2^3-3, 7=2^2+3=2^3-1)[/CODE]
I'd guess that such twin pairs are finite and fully listed there, even if there are infinite Mersenne, Fermat, and twin primes. AFAICT from a quick googling, [URL="http://www.mail-archive.com/mersenne@base.com/msg00865.html"]the last time[/URL] someone looked for Mersenne Twin Primes was in 1999, when the highest p known to make 2^p-1 prime was 3021377.

Mini-Geek 2011-01-20 16:33

I've started testing for twins (PRP or provable) of all primes on the top 5000 list that do [I]not[/I] have a base of 2 (since I can test those by downloading Karsten's lists). The timestamp for the list was "Thu Jan 20 04:51:06 CST 2011". I made a Python script to parse it out, which I'll post when I have results to post. Due to the vastly varying bases, GFNs, Phi's, and factorials/primorials, I don't see how I can really presieve this efficiently, so I'm just running it in PFGW with -f.

gd_barnes 2011-01-21 09:03

An excellent and very interesting work Tim. Nice job! :smile:

gd_barnes 2011-01-21 09:28

David,

Just to be clear: Do you plan to primality prove 5789*2^15513+3 ?

Tim,

I updated the status in post 32 to reflect what has now been done.


Gary

Mini-Geek 2011-01-23 17:00

1 Attachment(s)
[QUOTE=Mini-Geek;247669]I've started testing for twins (PRP or provable) of all primes on the top 5000 list that do [I]not[/I] have a base of 2 (since I can test those by downloading Karsten's lists). The timestamp for the list was "Thu Jan 20 04:51:06 CST 2011". I made a Python script to parse it out, which I'll post when I have results to post. Due to the vastly varying bases, GFNs, Phi's, and factorials/primorials, I don't see how I can really presieve this efficiently, so I'm just running it in PFGW with -f.[/QUOTE]

Results and Python 3.x parsing script attached. I also searched for twins of Proth primes (k*2^n+1) with a k over 10000 since those aren't on Karsten's list yet, and presieved that with srsieve. No primes found. Not counting numbers divisible by 3, I factored or tested 566 numbers. The other ones were of the type that they should be included on one of Karsten's lists.

henryzz 2011-01-23 17:43

[QUOTE=gd_barnes;247879]David,

Just to be clear: Do you plan to primality prove 5789*2^15513+3 ?

Tim,

I updated the status in post 32 to reflect what has now been done.


Gary[/QUOTE]
I will test it. It will take a while. I will try to use multicores. We are in no hurry otherwise I wouldn't do it. It will eventually get finished. This number is small enough for me to eventually finish but large enough that I feel the need to use multicores.

henryzz 2011-01-24 19:39

It will be about 100 hours of processing for phase 1. 4% done now. Begining to think multithreading is a waste of time for this small a number. I am running 2 threads and thread one has had 5/6 of the successes so far.

Mini-Geek 2011-01-24 19:52

1 Attachment(s)
[QUOTE=Mini-Geek;248734]Results and Python 3.x parsing script attached.[/QUOTE]

I've updated the script to handle extra-long/multi-line lines correctly. It can now read the entire current top 5000 list without any errors or placing numbers in the "confused" file (at least, from rank 1 to 5000 - not sure what'll happen if it sees all the other things at the top and bottom). In case anyone's interested, I'm attaching the updated version here.

mart_r 2011-01-28 20:15

[QUOTE=Mini-Geek;247088]I don't know if a record list exists for non +-1 twin prime, or (probably the same) a pair of twin primes/PRPs, where at least one is only known to be a PRP. Given the obscurity of that, I'd guess my 5789*2^15513+1, +3 pair have a good chance of being that.
[/QUOTE]

Uhm... What about the twin

(20431926447260679*4001#*(205881*4001#+1)+210)*(205881*4001#−1)/35 +5, +7 (5132 digits)

?

Mini-Geek 2011-01-28 21:58

[QUOTE=mart_r;250156]Uhm... What about the twin

(20431926447260679*4001#*(205881*4001#+1)+210)*(205881*4001#−1)/35 +5, +7 (5132 digits)

?[/QUOTE]

I was not aware of these twin primes. Is the pair listed on a list anywhere, or did you just find the pair, or what? A google search can't find it, and it's not large enough to be on any list I can see. I can confirm with PFGW that they are primes. (the number between the primes is easily trial factored to at least 33.34%)
What about narrowing the definition to twins where at least one of the pair would (at time of discovery) best be proved by general methods like ECPP, (e.g. no N-1, N+1, or N-1/N+1 combined test is useful) whether they have been proven or are still PRP? Maybe it qualifies for that record! :smile: If not, I give up and admit it's not a terribly interesting twin, it just happens to be the largest I found in my search.

davar55 2011-01-29 00:18

[QUOTE=Mini-Geek;247632][CODE][URL="http://factordb.com/index.php?id=1100000000291801677"]1381*2^6512+3[/URL] and below Mini-Geek (done, certificates in DB)
[URL="http://factordb.com/index.php?id=1100000000291801683"]7027*2^13017-3[/URL] [unreserved]
[URL="http://factordb.com/index.php?id=1100000000291801687"]755*2^13474-3[/URL] [unreserved]
[URL="http://factordb.com/index.php?id=1100000000287449188"]5789*2^15513+3[/URL] henryzz
[/CODE]Done PRPing all candidates and proving all candidates I said I would. All results I saved (started a little after n=20K) along with all primes in pfgw.log or pfgw-prime.log according to current proven status are attached.

2^13466917-3 would have taken a little over two days, but I put it on two cores for most of it, so it took closer to one day. Because it wasn't sieved very well, (only to 5 billion, or about 2^32) Prime95 chose P-1 bounds that gave it a 20% chance of finding a factor. Unfortunately it did not find a factor, even with such generous bounds, so I had to test it. Alas, the largest known twin Mersenne prime (i.e. 2^p-1 and (2^p-3 or 2^p+1) are prime) is just p=5: 29 and 31.
Just for fun, here are all known primes that are twin Mersenne or Fermat primes:
[CODE]2^16+1, +3 (65537, 65539)
2^4+1, +3 (17, 19)
2^5-1, -3 (29, 31)
3, 5, and 7, by various formulas (3=2^1+1=2^2-1, 5=2^1+3=2^2+1=2^3-3, 7=2^2+3=2^3-1)[/CODE]I'd guess that such twin pairs are finite and fully listed there, even if there are infinite Mersenne, Fermat, and twin primes. AFAICT from a quick googling, [URL="http://www.mail-archive.com/mersenne@base.com/msg00865.html"]the last time[/URL] someone looked for Mersenne Twin Primes was in 1999, when the highest p known to make 2^p-1 prime was 3021377.[/QUOTE]

So has anyone factored

[quote]As you can see, there are only two numbers (2^4253-3 and 2^11213-3) in the table that I have not been able to factor.
[/quote]either M(4253)-2 or M(11213)-2 since then?

Mini-Geek 2011-01-29 03:30

[QUOTE=davar55;250202]So has anyone factored

either M(4253)-2 or M(11213)-2 since then?[/QUOTE]

Not as far as I know, but I don't know of any serious ECM attempts, just that old TF to 2^32. Here are the FactorDB entries for those:
[URL="http://factordb.com/index.php?query=2^4253-3"]http://factordb.com/index.php?query=2^4253-3[/URL]
[URL="http://factordb.com/index.php?query=2^11213-3"]http://factordb.com/index.php?query=2^11213-3[/URL]

davar55 2011-01-30 18:00

[QUOTE=Mini-Geek;250222]Not as far as I know, but I don't know of any serious ECM attempts, just that old TF to 2^32. Here are the FactorDB entries for those:
[URL="http://factordb.com/index.php?query=2%5E4253-3"]http://factordb.com/index.php?query=2^4253-3[/URL]
[URL="http://factordb.com/index.php?query=2%5E11213-3"]http://factordb.com/index.php?query=2^11213-3[/URL][/QUOTE]

I started a db as a new thread in Data, M-2 something or other.

mart_r 2011-02-03 21:04

[QUOTE=Mini-Geek;250181]I was not aware of these twin primes. Is the pair listed on a list anywhere, or did you just find the pair, or what? A google search can't find it, and it's not large enough to be on any list I can see. I can confirm with PFGW that they are primes. (the number between the primes is easily trial factored to at least 33.34%)
[/QUOTE]

Hehe, I wondered if I should tell where I "found" it, but I was interested in whether you'd find out by yourself...
It's from the largest known prime triplet that doesn't contain a ±1 twin, found in March 2006 by Ken Davis.

henryzz 2011-02-07 19:54

[QUOTE=henryzz;249007]It will be about 100 hours of processing for phase 1. 4% done now. Begining to think multithreading is a waste of time for this small a number. I am running 2 threads and thread one has had 5/6 of the successes so far.[/QUOTE]
100 hours was a rubbish estimate based on lucky findings. 350 hours seems more likely now. I am now at test 43 14575/15526.

henryzz 2011-02-25 20:36

Thought I would post an update.:smile: Now at Test 222 10226/15526 which means 21% left of phase 1. Not long now.:smile:

henryzz 2011-03-05 13:23

[QUOTE=henryzz;253741]Thought I would post an update.:smile: Now at Test 222 10226/15526 which means 21% left of phase 1. Not long now.:smile:[/QUOTE]
Phase 1 finished today.

henryzz 2011-03-26 20:28

5789*2^15513+3 is proven prime after 228h 47mn 25s
The proof is at [url]http://www.sendspace.com/file/peur1w[/url]
Could someone please do a verify run on another pc preferably using the alternative verification software which is somewhere in the five or bust forum.

Mini-Geek 2011-03-26 22:09

[QUOTE=henryzz;256693]5789*2^15513+3 is proven prime after 228h 47mn 25s
The proof is at [url]http://www.sendspace.com/file/peur1w[/url]
Could someone please do a verify run on another pc preferably using the alternative verification software which is somewhere in the five or bust forum.[/QUOTE]

[URL="http://factordb.com/index.php?id=1100000000287449188"]FactorDB[/URL] is now processing your certificate (I uploaded it to it). It does not use Primo, but the alternative verification software you mention.
Primo 3.0.9, running on my box, says your certificate signature is valid.
Thanks for running this verification. :tu:

Mathew 2011-07-14 23:11

[URL="http://factordb.com/index.php?query=2%5E13017*7027-3"]7027*2^13017-3[/URL] is currently being processed.

Mathew 2011-09-12 06:27

[URL="http://factordb.com/index.php?id=1100000000291801687"]755*2^13474-3[/URL] is currently being processed


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