|
381*18^24108+1 is a probable prime.
So, I still have k = 122 base 18 |
[QUOTE=robert44444uk;95364]Base 9:
Covering set every 6n for [5,7,13,73]. Alternative covering set every 8 n for [5,41,17,193]. Lowest mooted Sierpinski is 2344 (k=439 is not Sierpinski because the k is also trivial). Lowest conjectured Riesel is 74, so should be easy to prove, but 4,16,36,64 are proving pesky. Note 16 and 64 are subsets of 4. [/QUOTE] 4*9^n-1 = (2*3^n-1)(2*3^n+1), so there are no primes to be found here. Same applies to 16,36,64. |
Algebra and base 9
Aiaia, such basic maths!! Why did I not spot that?
So all of these are trivial and the mooted Riesel base 9 is therefore proven. |
Speaking of basic maths, tell me if I'm right or wrong:
[quote]Base 8: Covering set [3,5,13] covering every 4 n. The corresponding Sierpinski number is 47, but it is not proven for the small fact that k=1 is known not to have small primes. (Think about it: 8^n+1= 2^3n+1[/quote] For 2^n+1 to be prime n has to be 2^m for some m. If 2^n+1 has to have n be a power of 2, there's no way, in 2^3n+1 to find an n value that makes 3n a power of 2. Did I miss something? |
For base 16 Does this work sierpinski number =27473
It has multiple covering sets. 2158*16^n+1 2857*16^n+1 2908*16^n+1 3061*16^n+1 4885*16^n+1 5886*16^n+1 6348*16^n+1 6663*16^n+1 6712*16^n+1 7212*16^n+1 7258*16^n+1 7615*16^n+1 7651*16^n+1 7773*16^n+1 8025*16^n+1 10183*16^n+1 10425*16^n+1 10947*16^n+1 12243*16^n+1 12900*16^n+1 13023*16^n+1 13438*16^n+1 14026*16^n+1 14661*16^n+1 14910*16^n+1 15370*16^n+1 15441*16^n+1 16015*16^n+1 16390*16^n+1 16846*16^n+1 17118*16^n+1 17970*16^n+1 18598*16^n+1 18828*16^n+1 19122*16^n+1 19465*16^n+1 19575*16^n+1 19668*16^n+1 19687*16^n+1 19725*16^n+1 20212*16^n+1 20446*16^n+1 20452*16^n+1 21115*16^n+1 21181*16^n+1 21436*16^n+1 21720*16^n+1 21943*16^n+1 22458*16^n+1 22747*16^n+1 23451*16^n+1 23682*16^n+1 24262*16^n+1 24505*16^n+1 24582*16^n+1 24790*16^n+1 26017*16^n+1 26215*16^n+1 26892*16^n+1 26977*16^n+1 These have been tested to n=4000. (under 27473 only) |
Base 16 Sierpinski
Extended above to 4400. The following primes were found. Stopping here. The numbers are free to take.:smile:
22747*2^16432+1 is prime! 12900*2^16508+1 is prime! |
Taking the following numbers:
2158*16^n+1 2857*16^n+1 2908*16^n+1 3061*16^n+1 4885*16^n+1 |
26977*2^20204+1 is prime! Time: 2.663 sec.
so status is [CODE]2158*16^n+1 jasong 2857*16^n+1 jasong 2908*16^n+1 jasong 3061*16^n+1 jasong 4885*16^n+1 jasong 5886*16^n+1 tnerual 6348*16^n+1 6663*16^n+1 6712*16^n+1 7212*16^n+1 7258*16^n+1 7615*16^n+1 7651*16^n+1 7773*16^n+1 8025*16^n+1 10183*16^n+1 10425*16^n+1 10947*16^n+1 12243*16^n+1 13023*16^n+1 13438*16^n+1 14026*16^n+1 14661*16^n+1 14910*16^n+1 15370*16^n+1 15441*16^n+1 16015*16^n+1 16390*16^n+1 16846*16^n+1 17118*16^n+1 17970*16^n+1 18598*16^n+1 18828*16^n+1 19122*16^n+1 19465*16^n+1 19575*16^n+1 19668*16^n+1 19687*16^n+1 19725*16^n+1 20212*16^n+1 20446*16^n+1 20452*16^n+1 21115*16^n+1 21181*16^n+1 21436*16^n+1 21720*16^n+1 21943*16^n+1 22458*16^n+1 23451*16^n+1 23682*16^n+1 24262*16^n+1 24505*16^n+1 24582*16^n+1 24790*16^n+1 26017*16^n+1 26215*16^n+1 26892*16^n+1 [/CODE] |
Base 16 Sierpinski
[QUOTE=Citrix;96821]For base 16 Does this work sierpinski number =27473
It has multiple covering sets. [/QUOTE] Citrix, I had considered 27473 with covering set [7,13,17,241] but sadly it is a trivial result (all n divided by 3) as the results for n=1..6 show 1 3^2*13^2*17^2 2 3*7*179*1871 3 3*17*23^2*43*97 4 3^2*11*13*1398967 5 3*7*17*2113*38189 6 3*241*401*1589803 So I will stick to my guns and I think 66741 is the smallest. The good news is that the work you have carried out has not gone to waste, you just need to check more k !! |
5886*2^108040+1 is prime! Time: 66.218 sec.
so status for base 16, sierpinski is ( with robert's remark) [CODE]2158*16^n+1 jasong 2857*16^n+1 jasong 2908*16^n+1 jasong 3061*16^n+1 jasong 4885*16^n+1 jasong 6348*16^n+1 6663*16^n+1 6712*16^n+1 7212*16^n+1 7258*16^n+1 7615*16^n+1 7651*16^n+1 7773*16^n+1 8025*16^n+1 10183*16^n+1 10425*16^n+1 10947*16^n+1 12243*16^n+1 13023*16^n+1 13438*16^n+1 14026*16^n+1 14661*16^n+1 14910*16^n+1 15370*16^n+1 15441*16^n+1 16015*16^n+1 16390*16^n+1 16846*16^n+1 17118*16^n+1 17970*16^n+1 18598*16^n+1 18828*16^n+1 19122*16^n+1 19465*16^n+1 19575*16^n+1 19668*16^n+1 19687*16^n+1 19725*16^n+1 20212*16^n+1 20446*16^n+1 20452*16^n+1 21115*16^n+1 21181*16^n+1 21436*16^n+1 21720*16^n+1 21943*16^n+1 22458*16^n+1 23451*16^n+1 23682*16^n+1 24262*16^n+1 24505*16^n+1 24582*16^n+1 24790*16^n+1 26017*16^n+1 26215*16^n+1 26892*16^n+1 and 27473 to 66740 [/CODE] |
i will take the base 16 sierpinski from 27473 to 66740 ... :sick:
|
[QUOTE=tnerual;96946]i will take the base 16 sierpinski from 27473 to 66740 ... :sick:[/QUOTE]
i'm not able to manage it. if someone has an app running under windows (or command line), able to do the job i will be happy :smile: the app has to do this: 1. test all k for n=2 2. remove all k with primes found in 1. 3. test all remaining k for n=3 4. remove all k with primes found in 3. and so on. i am totaly unable to program anything and excel, supposed to be my friend is not in reality ... maybe a programmer guru can do that ... it will help a lot of people (at least one :geek: ) |
[QUOTE=tnerual;96996]i'm not able to manage it.
if someone has an app running under windows (or command line), able to do the job i will be happy :smile: the app has to do this: 1. test all k for n=2 2. remove all k with primes found in 1. 3. test all remaining k for n=3 4. remove all k with primes found in 3. and so on. i am totaly unable to program anything and excel, supposed to be my friend is not in reality ... maybe a programmer guru can do that ... it will help a lot of people (at least one :geek: )[/QUOTE] You're worrying for nothing, dude. 16 is 2^4, and LLR notices this. Just sieve in base-16 and send it directly to LLR. When the LLR program sees base-16, it changes the base to 2 and multiplies the n-value by 4. No work needs to be done on the file, LLR is smart enough to figure it out on it's own. Edit: By the way guys: 2857*16^5478+1 is prime 2158*16^10906+1 is prime 4885 tested to n=50,000(base-16) no primes 3061 tested to n=50000(base-16) no primes 2908 tested to n=38000(base-16) no primes I'm unreserving all my numbers. Thanks. |
Base 16
Ok, here are the next few candidates for 16, all tested to n=4295. Last k tested was 47805
27592 27838 28593 28918 29445 30250 30397 31347 31912 32161 32350 32556 32673 33661 33771 34528 34543 35257 35320 35548 35818 35845 36735 37372 38440 38562 38776 39322 39337 39781 40000 40410 41530 42052 42376 42717 42745 43368 43398 43468 43485 44035 44131 44490 45306 45712 46471 46528 47176 47298 47395 47482 47616 |
Base 16 Sierpinski
And here is the last batch of Sierpinski 16, tested to n=5098
47818 48697 48772 48976 49386 49860 50166 50863 50865 51171 51427 52072 53226 53653 53941 54610 55306 55611 55846 55897 56866 57238 57795 57867 58582 58791 58855 59890 60070 60343 60541 60891 61617 61687 62802 63390 63405 63411 63853 64518 64620 65077 65127 65397 65536 square 65623 |
In post 112, k=40000 is square
|
[QUOTE=robert44444uk;97130]In post 112, k=40000 is square[/QUOTE]
You did not exclude numbers divisible by 16:grin: |
Sierpinski 16
I will reserve those above k=60000 and take them quite high
|
[QUOTE=Citrix;97131]You did not exclude numbers divisible by 16:grin:[/QUOTE]
True, can you spot them and weed them out |
Base 16 Sierpinski
Doing a quick check through the top 5000 site
42717*2^905792+1 is prime! L159 2005 44131*2^995972+1 is prime! SB3 2002 - very valuable "Seventeen or Bust", a first prime for 44131 and so this will save man years of work |
Base 18 Sierpinski
Unreserving Base18 Sierpinski
k=122 tested till n=110746 no primes found - anyone else want to try? 381*18^24108+1 is a probable prime. Do I have to register this prime somewhere? Prime Pages says, it is too small. k=18 and k=324 stopped testing, because k = base |
[quote=Xentar;97217]
381*18^24108+1 is a probable prime. Do I have to register this prime somewhere? Prime Pages says, it is too small. [/quote] Simply no. |
Base 12 Sierpinski
Prof Caldwell has pointed out an error in the k value quoted for Base 12 Sierpinski, due to me choosing the wrong covering set to analyse.
In fact the covering set 5,13,29 produces a Sierpinski at k=521, and nominal sieving leaves 261,378 and 404 to find primes for, in addition to the GFN k's 12 and 144. Recalculating gives the Riesel at 376, with the following unfound at n=5000 25,27,64,300,324 which are all highly composite (all 2,3, and 5), I will leave it to others to demonstrate that these factor (I am feeling lazy today) |
Base 16 Sierpinski
I looked at the base 16 sierpinski for all k >60000, report as follows:
k n prime 60343 5745 60541 44085 60891 10036 61617 7845 61687 8948 62802 42004 63390 23511 63411 18016 64518 36998 65127 5206 65397 6222 No primes were found for the following k, tested up to the following n: 60070 53562 63405 42335 63853 39256 64620 75696 65077 58138 65623 64630 Will reserve all between k=50000 and k=60000 |
Is there somewhere where we can find the n-values for the low k? If not, maybe someone could help with scripts(in Windows) that can take a list of prime k/n equations and find the k-values that are missing below a certain level. I could do each base up to 10,000 then and there would be a database for n-values that yield certain k/b prime and a list of what's left after 10,000 as well as a reference to this thread.
If there isn't a reference site for this, maybe someone could help me accomplish this? |
[quote=robert44444uk;95436]After a bit of fiddling about with [7,43,37,31,13] came up with the riesel candidate 213410 for base 6. 133946 is trivial.
Regards Robert Smith[/quote] [quote=robert44444uk;95426]Did a little work this afternoon of base 6. Using the covering set [7,43,37,31,13] repeating every 24n provides a Sierpinski number 243417. I will try to do the Riesel later. Note that the alternative set [7,43,37,31,97] repeating every 24n could also provide a lower Sierpinski value. However 243417 is at 0.73% of the product of this set's cover primes, and there are only 24 values to check, if I was any good at statistics I could tell you what the probability is, but I am not!! Regards Robert Smith[/quote] According to my calculations: Here are the lowest Riesel and Sierpinski numbers base 6: Riesel: k=84687 Sierpinksi: k=174308 Both have a covering set of [7,13,31,37,97] and repeat every 12 n. I realize the above are older posts from Jan. 2007 and that perhaps someone has come up with some updated info. since that time. If there is newer info. that I missed reference base 6, can you refer me to it? Or if not and you'd like to know how I came up with the values, I can explain. Maybe this will make the conjectures a little easier to prove base 6! :smile: I am in the process of putting together all of the information that I can in order to create a web page that has all of the Riesel-Siepinski information for all bases <= 50. So any information that I can get helps. Thanks, Gary |
Was base 6 to 18 discussion moved ?
I'm looking for a continuation of the base 6 to 18 thread discussion that seemed to end abruptly in Feb. Was it moved somewhere or did it just die out?
There was a lot of interesting work going on in the thread and I am putting together a web page of much of the information that is in it. Thanks, Gary |
It's [url=http://www.mersenneforum.org/showthread.php?t=6895]right here[/url], just two threads down from yours at the moment. :smile: Maybe you passed over it somehow?
|
[quote=Anonymous;119318]It's [URL="http://www.mersenneforum.org/showthread.php?t=6895"]right here[/URL], just two threads down from yours at the moment. :smile: Maybe you passed over it somehow?[/quote]
No...of course I saw that thread. I said was the CONTINUATION of it moved? That thread just died in Feb. 2007. I don't get it. There was so much information flying back and forth and then it's like it went off into never-never land somewhere like a moderator moved it somewhere or something. I guess another question would be...is there more info. for the Riesel-Sierpinski problems for base 6 and higher other than just that thread? Gary |
I think people just lost interest. Base 10 is still alive. You can use the other projects section, if you want to work on a particular base.
|
merged gd_barnes thread questioning the activity level of this thread....
|
[quote=gd_barnes;119322]No...of course I saw that thread. I said was the CONTINUATION of it moved? That thread just died in Feb. 2007. I don't get it. There was so much information flying back and forth and then it's like it went off into never-never land somewhere like a moderator moved it somewhere or something.
I guess another question would be...is there more info. for the Riesel-Sierpinski problems for base 6 and higher other than just that thread? Gary[/quote] Oh, I see now. :smile: I had misunderstood your post. :blush: |
[quote=Citrix;119330]I think people just lost interest. Base 10 is still alive. You can use the other projects section, if you want to work on a particular base.[/quote]
Oh, that's too bad. I had hoped it 'continued' somewhere else. It looks like Robert mostly kept it going but he left and it was getting hard to follow without a single place for all of the info. by base. Maybe those things caused it to die out. I have a web page that lists all known Riesel primes base 10 at [URL="http://gbarnes017.googlepages.com/primes-kx10n-1.htm"]gbarnes017.googlepages.com/primes-kx10n-1.htm[/URL] including the info. in this thread and the other one in this forum. Masser, would you have any objection to me seeing if I can get things started again in this thread? I've got a spreadsheet started that contains a lot of additional research that I've done that I'll be using to create a web page in the near future. Perhaps a web page will jump start interest again. More than one person had asked for it previously in this thread. Thanks, Gary |
[QUOTE=gd_barnes;119369]
Masser, would you have any objection to me seeing if I can get things started again in this thread? [/QUOTE] I have no objection at all, but perhaps all of these threads about other bases belong in the new Open Projects sub-forum. As moderator of this project, I feel that we should maintain focus on the base 5 project. The other bases only serve as a distraction to our project. |
[quote=masser;119428]I have no objection at all, but perhaps all of these threads about other bases belong in the new Open Projects sub-forum. As moderator of this project, I feel that we should maintain focus on the base 5 project. The other bases only serve as a distraction to our project.[/quote]
That would probably be a good idea, considering that Sierpinski Base 4 is already over there anyway. |
[quote=masser;119428]I have no objection at all, but perhaps all of these threads about other bases belong in the new Open Projects sub-forum. As moderator of this project, I feel that we should maintain focus on the base 5 project. The other bases only serve as a distraction to our project.[/quote]
OK, thanks. Makes sense. I'll get something started over there in the near future. I have a large amount of info. put together about most bases <= 26 but I still have a fair amount of data collection to go from various sources. One of them being the Base 4 thread in Open Projects that Anon referred to. Gary |
If I may make a suggestion:
There are a lot of bases being covered, and I'm not sure how many threads there are going to be. Maybe there should be a reservation thread and a comments thread, with a moderator using the first post as the update area in the reservations thread. What do you think? Edit: and now I see it's already there. Sorry for clogging the thread. |
Web pages for bases 2-32 coming in next 1-2 weeks
[quote=jasong;119926]If I may make a suggestion:
There are a lot of bases being covered, and I'm not sure how many threads there are going to be. Maybe there should be a reservation thread and a comments thread, with a moderator using the first post as the update area in the reservations thread. What do you think? Edit: and now I see it's already there. Sorry for clogging the thread.[/quote] Sometime in the next 1-2 weeks, I will have a huge amount of information about all bases <= 32 put into 2-3 web pages and will start an entirely new thread here in 'open projects'. I'm still pulling info. from various sources and 'filling in some holes' on a few bases by doing some testing here and there. After doing a lot of additional searching myself, I've proven several bases not previously proven in this thread and gotten it down to where there are currently 2 Riesel and 6 Serpinski bases <= 32 that have only ONE remaining stubborn k that needs a prime. After additional searching, I'll be asking for help with those one-k bases remaining, on some bases that have many remaining candidates that need a prime, and on starting some bases that have had no work done yet. I'll put in some info. on one of the pages about reservations. The Riesel and Serpinski conjectures make for most interesting Prime Search projects in all bases and bringing all of the info. together for multiple bases has both been very interesting and a great learning experience! Gary |
Proof of Riesel conjecture base 12
Here is a proof of the Riesel conjecture base 12 that was previously analyzed and searched in this thread:
Conjecture: The Riesel k = 376 with a covering set of 5, 13, 29. This was already given here. k's where k==1 mod 11 are eliminated with a trivial factor of 11. There are 3 k's remaining where a prime has not been found. They are k=25, 27, and 64. Before proving this, for everyone's "entertainment" :smile:, here are the top-10 k's with the largest first primes found: k / (n) 157 (285) 46 (194) 259 (40) 304 (40) 94 (36) 292 (30) 147 (28) 301 (27) 349 (25) 58 (23) "My conjecture": k=25, 27, and 64 are composite for all n but do not have a specific covering set of numeric factors. Proof of "My conjecture" for k=25 and 64 generalized for all possible values of k: For all k=m^2 and k==12 mod 13 (both must be true), the following algebraic and numeric factors are present for all n: 1. For all odd n, there is a factor of 13. 2. For all even n, let k=m^2 and let n=2*q. There are now algebraic factors of (m*12^q - 1) * (m*12^q + 1). Therefore 25*12^n-1 and 64*12^n-1 must be composite for all n. Proof of "My conjecture" for k=27 generalized for all possible values of k: For all k=3*m^2 and k==1 mod 13 (both must be true), the following algebraic and numeric factors are present for all n: 1. For all even n, there is a factor of 13. 2. For all odd n, let k=3*m^2 and let n=2*q-1. There are now algebraic factors of [m*3^q*2^(2q-1) - 1] * [m*3^q*2^(2q-1) + 1]. Therefore 27*12^n-1 must be composite for all n. This of course begs the question: What really is the Riesel number base 12? IMHO, it is still k=376 (not k=25) and the above examples are the equivalent of trivial k's base 12, i.e. k=1, 12, 23, 34, etc., and just needed to be proven as such. But I'm open to hearing anything on the matter. It also begs the question of more specifically defining a covering set. I think I've demonstrated here that a k can have both a partial covering set of 'numeric' factors as well as a partial covering set of algebraic factors such that it becomes a full covering set. But once again, IMHO I would think we'd want only "numeric" (not algebraic) covering sets for the conjectures otherwise we'd wind up with very low k's for the conjectures for many bases such as this one. Any other thoughts, comments, corrections, and opinions are also welcome. Gary |
[quote=jasong;97017]
Edit: By the way guys: 2857*16^5478+1 is prime 2158*16^10906+1 is prime 4885 tested to n=50,000(base-16) no primes 3061 tested to n=50000(base-16) no primes 2908 tested to n=38000(base-16) no primes I'm unreserving all my numbers. Thanks.[/quote] Jasong, I'm doing some double-checking before posting my web pages... Neither one of these numbers is prime. 2857*16^5478+1 has a factor of 11. It looks like you may have tested base 2 because 2857*2^5478+1 is prime. But since n is not a multiple of 4, it doesn't help base 16. 2158*16^10906+1 also has a factor of 11. 2158*2^10906+1 has a factor of 127. So I'm not sure what you tested there. I show both k's as composite to n=6580 and will be testing all k's above n=10K for base 16 before publishing the pages. If you have a prime on one of these k's, please let me know. Thanks, Gary |
[QUOTE=gd_barnes;120479]Jasong,
I'm doing some double-checking before posting my web pages... Neither one of these numbers is prime.[/QUOTE] Thanks for the double check, I have no idea what went wrong. |
[quote=jasong;120573]Thanks for the double check, I have no idea what went wrong.[/quote]
No problem...I found the primes after a little more searching: 2857*16^6832+1 is prime 2158*16^10905+1 is prime And one more from your other k's: :wink: 3061*16^8322+1 is prime All k's below the Sierpinski k=66741 for base 16 are now searched to n=13K. 77 k's are remaining after eliminating all k's with higher primes found by prior projects on bases 2 and 4...not too bad for such a low search range. (3 of the 77 are in effect still being searched by those projects.) My web pages with most known Riesel/Sierpinski conjecture info. bases 2-32 are complete. Check for a new thread here on Thursday. I'll have some sieved files for several different bases ready to be handed out and searched. Gary |
Report future status at "Conjectures 'R Us"
All bases 6 to 18 searchers,
All conjectures for bases > 2 except those being worked by other major projects are now being coordinated in the new "Conjectures 'R Us" effort in this Open Projects forum. Please report all future reservations and statuses for bases 6 to 18 in the reservations/statuses thread for that effort. Web pages have been created that show all current relavent info. After a couple of days, I'll request that this thread be locked to avoid any duplication of future effort. Thanks, Gary |
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