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 2007-01-10, 09:53 #1 Citrix     Jun 2003 32×52×7 Posts Sierpinski and Riesel number (Fixed k, Variable base) Has there been any work done to find S/R for fixed k and variable base? eg. What is the lowest base=b for k=2 such that 2*b^n+1 is never prime? What k's can never be sierpinski numbers? Is there any proof to this. How does one generate the sierpinski base for a given k? (The same questions for Riesel side)
2007-01-10, 12:32   #2
thommy

Dec 2006

418 Posts

Quote:
 Originally Posted by Citrix What is the lowest base=b for k=2 such that 2*b^n+1 is never prime?

b=4, as 2*1^1+1 and 2*2^1+1 and 2*3^1+1 are prime
For all n : 2*4^n+1=2*1^n+1=2*1+1=3=0 (mod 3), so every number divisible by 3.
Those questions seem not that interesting.

 2007-01-10, 15:08 #3 robert44444uk     Jun 2003 Oxford, UK 5×379 Posts The alternative Sierpinski/ Riesel works off b^n+/-k. Numbers of this form have the same properties as k*b^n+/-1. (trust me, this is the case!!) It is not so popular because you cannot prove the numbers prime, only prp. But there is a whole community of people out there interested in finding just that... numbers that are prp but not prime. Check http://www.primenumbers.net/prptop/prptop.php Henri lists the top 10000 prps and therefore it is easy to get into this list. But first you need to work out the probable Sierpinski/ Riesels for b^n+/-k, and then look at eliminating all k up to the chosen value. In this way you will find prps of some other k value such as L in b^n+/-L which Henri will be pleased to list
2007-01-10, 17:36   #4
Citrix

Jun 2003

32×52×7 Posts

Quote:
 Originally Posted by thommy b=4, as 2*1^1+1 and 2*2^1+1 and 2*3^1+1 are prime For all n : 2*4^n+1=2*1^n+1=2*1+1=3=0 (mod 3), so every number divisible by 3. Those questions seem not that interesting.
Thommy.
2 is not considered a sierpinski number for base 4, since the solution is trivial and no covering set is involved.

 2007-01-23, 04:55 #5 Citrix     Jun 2003 32×52×7 Posts For k=2 base=512 will never produce a prime! (2*512^n+1) The following numbers below it remain The following values remain. 38 101 104 122 167 206 218 236 257 263 287 305 353 365 368 383 395 416 461 467 497 Will try to eliminate some.
 2007-01-23, 10:47 #6 Citrix     Jun 2003 32×52×7 Posts Found another lowest number =1307. 512 is a trivial solution, this is not. All below checked to n=1000. 101 -->done to 4500 167 206 218 236 257 287 305 353 365 368 383 395 416 461 467 497 512 --> Can be removed trivially 518 542 578 626 635 647 695 698 752 758 764 773 788 801 812 836 842 867 869 878 887 899 908 914 917 932 947 948 954 992 1004 1052 1058 1073 1079 1082 1097 1112 1139 1142 1187 1193 1232 1262 1277 1286 Primes 2*104^1233+1 2*122^755+1 2*263^957+1 2*38^2729+1 2*821^945+1 2*845^877+1 2*926^765+1 2*968^917+1 2*1022^727+1 2*1028^669+1 2*1181^789+1 2*1253^697+1 2*1283^765+1 Will continue to prove 1307 is the smallest such number. Have not found a -1 number upto 250,000. Not sure if there is one. may be the same covering set as +1 can be used. Need help here, if anyone can offer. Last fiddled with by Citrix on 2007-01-23 at 11:44
 2007-01-24, 05:16 #7 Citrix     Jun 2003 157510 Posts Here is the updated list. I would like to share the numbers with everyone, I am only working on a few, the rest are available. Could the moderators keep this thread clean. Code: base n weight reserved by 101 4500 903 167 4000 235 206 4000 614 218 4000 465 236 4000 497 257 4000 187 Citrix 287 4000 260 305 4000 1049 365 4000 616 368 4000 379 383 10000 76 Citrix 461 4000 535 467 4000 288 518 4000 227 542 2500 158 Citrix 578 2500 472 626 2500 519 635 2500 669 647 2500 370 695 2500 655 752 2500 169 Citrix 758 2500 422 773 10000 83 Citrix 788 2500 665 801 2500 1440 836 2500 831 869 2500 818 878 2500 435 887 2500 495 899 2500 449 908 2500 451 914 2500 982 917 2500 297 932 2500 693 947 2500 547 954 3000 1697 1004 2000 394 1052 2000 232 1058 2000 606 1073 2000 413 1079 2000 631 1082 2000 606 1097 2000 407 1139 2000 567 1142 2000 370 1187 2000 362 1193 2000 311 1232 2000 528 1262 2000 372 1277 2000 187 Citrix 1286 2000 721
 2007-01-25, 08:48 #8 Citrix     Jun 2003 32·52·7 Posts Some recent primes!!! 2*497^1339+1 2*698^1885+1 2*764^1189+1 2*812^1003+1 2*842^1919+1 2*867^1280+1 2*867^1367+1 2*867^1856+1 2*948^1242+1 2*992^1179+1 2*1112^1091+1 2*353^2313+1 2*395^2625+1 2*416^2517+1 2*518^4453+1 2*635^2535+1 2*635^2937+1 2*1187^2907+1 2*1262^2575+1 2*1286^2145+1 Code: 101 5000 903 167 5000 235 206 5000 614 218 5000 465 236 5000 497 257 5000 187 Citrix 287 5000 260 305 5000 1049 365 5000 616 368 5000 379 383 10000 76 Citrix 461 5000 535 467 5000 288 542 5000 158 Citrix 578 3000 472 626 3000 519 647 3000 370 695 3000 655 752 5000 169 Citrix 758 3000 422 773 10000 83 Citrix 788 3000 665 801 3000 1440 836 3000 831 869 3000 818 878 3000 435 887 3000 495 899 3000 449 908 3000 451 914 3000 982 917 3000 297 932 3000 693 947 3000 547 954 5000 1697 1004 3000 394 1052 3000 232 1058 3000 606 1073 3000 413 1079 3000 631 1082 3000 606 1097 3000 407 1139 3000 567 1142 3000 370 1193 3000 311 1232 3000 528 1277 5000 187 Citrix Average wt=521.826087 Total wt=24004
2007-01-25, 12:35   #9
robert44444uk

Jun 2003
Oxford, UK

5×379 Posts

Quote:
 Originally Posted by Citrix Found another lowest number =1307. 512 is a trivial solution, this is not. Will continue to prove 1307 is the smallest such number. Have not found a -1 number upto 250,000. Not sure if there is one. may be the same covering set as +1 can be used. Need help here, if anyone can offer.
Citrix, what is yor covering set for 1307? Obvioulsy there are not many n for which small factors cannot be found but there are 708 n values in the first 100,000 n which have no factors smaller than 50 million.

For example, my NewPgen file reads (for the "Sierpinski":

51763650:P:0:1307:257
2 123
2 387
2 435
2 723
2 891
2 1131
2 1155
2 1443
2 1491
2 1515
2 1803
2 1947
2 1971
2 1995.....

None of these are prime up to n=4731

Last fiddled with by robert44444uk on 2007-01-25 at 12:36

2007-01-25, 19:02   #10
Citrix

Jun 2003

32×52×7 Posts

Quote:
 Originally Posted by robert44444uk Citrix, what is yor covering set for 1307? Obvioulsy there are not many n for which small factors cannot be found but there are 708 n values in the first 100,000 n which have no factors smaller than 50 million. For example, my NewPgen file reads (for the "Sierpinski": 51763650:P:0:1307:257 2 123 2 387 2 435 2 723 2 891 2 1131 2 1155 2 1443 2 1491 2 1515 2 1803 2 1947 2 1971 2 1995..... None of these are prime up to n=4731
You are correct. Some error occured on my end. Thanks for pointing it out. But when I tried use Srsieve, it said all the numbers were eliminated. So I assumed it was a Sierpinksi number of this type. Though now when I run Srsieve it says some numbers are left.

I will stick to values under 512 then. I don't think that 2 can be a sierpinki/riesel number for any base. Nor can any of the low k values.
Code:

101	5000	903
167	5000	235
206	5000	614
218	5000	465
236	5000	497
257	5000	187	Citrix
287	5000	260
305	5000	1049
365	5000	616
368	5000	379
383	10000	76	Citrix
461	5000	535
467	5000	288
So if someone was to plot the sierpinski numbers (Y axis) and use the count (x axis) does the slope of the curve eventually become almost 0. If yes then it means that low k values are more likely to produce primes than high k values. Does anyone have enough data to plot this. Any thoughts on why low k's like 2, 3 can never be sierpinski numbers to any base...

Thanks!

2007-01-26, 00:31   #11
geoff

Mar 2003
New Zealand

22058 Posts

Quote:
 Originally Posted by Citrix But when I tried use Srsieve, it said all the numbers were eliminated. So I assumed it was a Sierpinksi number of this type. Though now when I run Srsieve it says some numbers are left.
There was a bug, hopefully fixed in version 0.6.4, that could cause this problem.

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