Sierpinski and Riesel number (Fixed k, Variable base)

[Citrix]

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)

[thommy]

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.

[robert44444uk]

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

[Citrix]

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.

[Citrix]

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.

[Citrix]

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.

[Citrix]

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.

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

[Citrix]

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

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

[robert44444uk]

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

[Citrix]

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.

 
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!

[geoff]

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.