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-   -   Sierpinski and Riesel number (Fixed k, Variable base) (https://www.mersenneforum.org/showthread.php?t=6918)

Citrix 2007-01-10 09:53

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)

thommy 2007-01-10 12:32

[quote=Citrix;95709]What is the lowest base=b for k=2 such that 2*b^n+1 is never prime?[/quote]


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 2007-01-10 15:08

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 [url]http://www.primenumbers.net/prptop/prptop.php[/url]

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 2007-01-10 17:36

[QUOTE=thommy;95721]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.[/QUOTE]

Thommy.
2 is not considered a sierpinski number for base 4, since the solution is trivial and no covering set is involved.

Citrix 2007-01-23 04:55

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 2007-01-23 10:47

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:xmastree:

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.:surrender

Citrix 2007-01-24 05:16

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.
:xmastree: :curtisc:
[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


[/code]

:help:

Citrix 2007-01-25 08:48

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


[/code]

robert44444uk 2007-01-25 12:35

[QUOTE=Citrix;96835]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.:surrender[/QUOTE]

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 2007-01-25 19:02

[QUOTE=robert44444uk;96970]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[/QUOTE]

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
[/code]

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!
:smile:

geoff 2007-01-26 00:31

[QUOTE=Citrix;96991]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.[/QUOTE]
There was a bug, hopefully fixed in version 0.6.4, that could cause this problem.


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