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Old 2007-01-05, 16:41   #1
robert44444uk
 
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Default Sierpinski/ Riesel bases 6 to 18

OK guys, I know I should not do this, but I thought I would post some results of my limited studies of Sierpinski/ Riesel series in bases other than 2,3,4 and 5. Some of you with efficient programming should be able to take this study further. Here in Bangladesh I am unable to access my Maple software, which would have speeded things up a lot.

Some of you might have seen recent Yahoo postings relating to my hypothesis that there is a covering set for every base for both Riesel and Sierpinski series. This is going to be difficult to prove as such, but certainly it seems very likely. I am likely to move my research to the special base form base=2^n-1, where I need to find a covering set in principle for all values and these may not be easy to find.

Anyway here is information on bases 6 to 18:

Base 6:

Will need a lot of work – a covering set, repeating every 12 n is [7,13,31,37,43]. Lowest Sierpinski and Reisel candidates not known. But neither can be greater than 4488211.

Another possible covering set, with repeat of 12 is [7,43,37,31,97]

Base 7:

Totally horrible. Possible covering set with repeat every 24 n is [19,5,43,1201,13,181,193,73], also 5 other sets perming 73, 193 and 409.

Sierpinski and Riesel numbers are both lower than 162643669672445

Work is needed to find a low k value which is Riesel or Sierpinski.

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

For Reisel k=112 looks the most likely candidate. A prime needs to be found however for k=14

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.

Base 10:

Covering set every 6n is [11,37,7,13]. Lowest known non trivial Sierpinski is 9351, lowest Riesel 10176. No work done yet to prove these.

Base 11:

Covering set every 6n is [3,7,19,37]. Lowest Sierpinski k is thought to be 1490. 3 ks need to be eliminated 416, 958, 1468

Lowest Riesel is thought to be 4624. 12k’s still need to be eliminated. 62, 682, 862, 904, 1528, 2410, 2690, 3110, 3544, 3788, 4208, 4564.

Base 12:

Covering set every 6k is [13,157,7,19]. Lowest Sierpinski mooted to be 14600 and lowest Riesel 16329. No work on this yet.

Base 13:

Covering set every 3 n, [7,17,5]. Lowest Sierpinski is proven to be 132. Lowest non trivial Riesel is thought to be 302, but need to prove k=288 is not.

Base 14:

With a covering set every 2n [3,5] this proves to be easy. Proven Sierpinski and Riesel values are both proven at 4.

Base 15:

Horrible. A covering set is [241,113,211,17,1489,13,3877], and Sierpinski and Riesel values are therefore less than 7330957703181619. As bad as the base 3 problem.

Base 16:

Sierpinski number not known, quite complex to calculate. Covering set [7,13,19,37,73] have 36 combinations to check Sadly the simple covering set 7,13,17 only appears in trivial solutions. On the Riesel side not so sad, conjectured to be 120. However n=9 is a problem, this number is well studied as 9.16^n-1 = 9.2^(4n)-1, and no values for n less than 207177

Base 17:

Covering set every 4 n, [3,5,29]. Sierpinski value of 278, the following candidates need a prime 88,92,160,244,262. On the Riesel side, the lowest is conjectured to be 86, with only k=44 needing to find a prime.

Base 18:

Covering set every 6n is [19,13,5]. Sierpinski value is 398. 4 k candidates seek primes 18,324,122,381. Riesel is proven to be 246

Any help anyone can give in taking this to higher bases, or proving the mooted Sierpinskis and Riesels is welcome.

Regards

Robert Smith

Last fiddled with by robert44444uk on 2007-01-05 at 16:45
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Old 2007-01-05, 18:32   #2
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For the fun of it i will do some work on Sierpinski Base 17 k=88.
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Old 2007-01-05, 18:41   #3
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i will give a small try at 1468*11^n+1 ...
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Old 2007-01-05, 18:55   #4
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You can use NewPGen to figure out some of the lowest Riesel/Sierpinski numbers. Use the "k*b^n+/1 with k fixed" sieves, and then use the "sieve until" option. Sieve up to p=1000 (or a million - something that doesn't take too long), and then have it update k and start again.

If this is done correctly, NewPGen will sieve k=1, 2, 3, and so on and create files 1.txt, 2.txt, 3.txt, etc. Sieve over a relatively small n set (1<=n<=2000) to keep the files small. Then just keep an eye on the size of the files - when you find one with size 0 - you've PROBABLY found a Riesel/Sierpinski number; do some fun math to figure out the covering set.

This method is sloppy, inefficient, etc.... but it's easy to set up and run. The alternative is to do some careful input file creation with PFGW or your own coding. There's also a tool (psieve is what I think it's called) out there for calculating the weights of certain sequences. I think it works for k*b^n+/-1, with bases other than 2. You can fix b and loop through many k values and then look for sequences with weight 0. Actually, now that I am thinking of it, this is probably the best way (without actually doing the math) to find the Sierpinski/Riesel numbers for other bases.

Hope this helps those of you interested...

best regards,
masser

Last fiddled with by masser on 2007-01-05 at 18:57
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Old 2007-01-05, 20:32   #5
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Robert, since you are posting the stuff for all bases, could you post the stuff for base 3 and well as base 4. I have been working on base 4 sierpinski with some 4 candidates left, I don't know about base 4 Riesel.

Thanks for the post.

edit:-Would be very nice to extend this list to higher bases of 2 eg(4,8,16,32...)since they are very easy to PRP

Last fiddled with by Citrix on 2007-01-05 at 20:51
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Old 2007-01-05, 20:35   #6
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Quote:
Originally Posted by masser View Post
You can use NewPGen to figure out some of the lowest Riesel/Sierpinski numbers. Use the "k*b^n+/1 with k fixed" sieves, and then use the "sieve until" option. Sieve up to p=1000 (or a million - something that doesn't take too long), and then have it update k and start again.

If this is done correctly, NewPGen will sieve k=1, 2, 3, and so on and create files 1.txt, 2.txt, 3.txt, etc. Sieve over a relatively small n set (1<=n<=2000) to keep the files small. Then just keep an eye on the size of the files - when you find one with size 0 - you've PROBABLY found a Riesel/Sierpinski number; do some fun math to figure out the covering set.

This method is sloppy, inefficient, etc.... but it's easy to set up and run. The alternative is to do some careful input file creation with PFGW or your own coding. There's also a tool (psieve is what I think it's called) out there for calculating the weights of certain sequences. I think it works for k*b^n+/-1, with bases other than 2. You can fix b and loop through many k values and then look for sequences with weight 0. Actually, now that I am thinking of it, this is probably the best way (without actually doing the math) to find the Sierpinski/Riesel numbers for other bases.

Hope this helps those of you interested...

best regards,
masser
You can also modify srsieve to do this and it would be much faster. It isn't a difficult change as I had done it a couple of months ago. I might still have the change somewhere on my hard-drive if someone is interested.
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Old 2007-01-05, 20:58   #7
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You don't have to modify srsieve actually. If a k is removed from the sieved it is recorded in the log file. So all you need to do is write a program to generate the input file for srsieve.
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Old 2007-01-05, 23:00   #8
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Quote:
Originally Posted by Citrix View Post
You don't have to modify srsieve actually. If a k is removed from the sieved it is recorded in the log file. So all you need to do is write a program to generate the input file for srsieve.
The point is that you only want to do a a thousand or more k at a time. Sieve to 1e5 (or whatever limit you want). If nothing pops out as a Riesel or Sierpinski number, then do the next 1000. Iterate until one is found.
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Old 2007-01-05, 23:02   #9
jasong
 
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Quote:
Originally Posted by masser View Post
You can use NewPGen to figure out some of the lowest Riesel/Sierpinski numbers. Use the "k*b^n+/1 with k fixed" sieves, and then use the "sieve until" option. Sieve up to p=1000 (or a million - something that doesn't take too long), and then have it update k and start again.

If this is done correctly, NewPGen will sieve k=1, 2, 3, and so on and create files 1.txt, 2.txt, 3.txt, etc. Sieve over a relatively small n set (1<=n<=2000) to keep the files small. Then just keep an eye on the size of the files - when you find one with size 0 - you've PROBABLY found a Riesel/Sierpinski number; do some fun math to figure out the covering set.

This method is sloppy, inefficient, etc.... but it's easy to set up and run. The alternative is to do some careful input file creation with PFGW or your own coding. There's also a tool (psieve is what I think it's called) out there for calculating the weights of certain sequences. I think it works for k*b^n+/-1, with bases other than 2. You can fix b and loop through many k values and then look for sequences with weight 0. Actually, now that I am thinking of it, this is probably the best way (without actually doing the math) to find the Sierpinski/Riesel numbers for other bases.

Hope this helps those of you interested...

best regards,
masser
I'm going to try masser's idea and see how it goes for base-6. Just to make it official:

Reserving base-6 to try to find Riesel number.

Edit: Possibly stupid question, is there a simple way to search all the newpgen files in a directory and find the one with no values?

Edit2: NewPGen allows increasing n, but not k, so I'm I don't think it's possible to implement masser's idea without programming knowledge. Would be happy to be proven wrong.

Edit3: Reserving 14*8^n-1

Last fiddled with by jasong on 2007-01-05 at 23:09
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Old 2007-01-05, 23:33   #10
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It seems very likely like 14*8^n-1 is composite for all n. NewPgen says it's composite for all n up to 50,000,000.

Edit: Reserving 92*17^n-1

Edit: 92*17^4-1 trivially factors prime!: 7683931

could there be typos in that message?

Edit: reserving 4*9^n-1(yes, I know, I'm editing like crazy.)

Last fiddled with by jasong on 2007-01-05 at 23:45
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Old 2007-01-06, 01:36   #11
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Quote:
Base 17:

Covering set every 4 n, [3,5,29]. Sierpinski value of 278, the following candidates need a prime 88,92,160,244,262. On the Riesel side, the lowest is conjectured to be 86, with only k=44 needing to find a prime.
Quote:
Originally Posted by jasong View Post
Edit: 92*17^4-1 trivially factors prime!: 7683931
A prime is needed for 92*17^n+1.
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