Exotic covers
 

I am trying to determine 3 things (in order of difficulty):

1. The lowest base with 11-cover
2. The lowest k for a base with 11-cover
3. The lowest base, if one exists that provides k with the minimum Sierpinski or Riesel k

The 1st problem can be solved by brute force by running bigcover.exe increasing the base by 1 and waiting for a base to produce 11 fresh primes at b^11-1. I had a near miss with b=4096 that produced 10. I am currently up to b=109794 and I expect by b=200000 I will have found it.

The second looks mind bending but an interesting exercise. The lowest k for a base might be provided by the 11 primes that are the lowest possible in b^11-1. Only certain primes are allowed and the lowest of these are:

p[1],p[2]....= 23,67,89,199,331,353,397,419,463,611,661

These each appear as factors of b^11-1 for 10 distinct mods of p[1],p[2]...viz

[CODE]23 67 89 199 331 353 397 419 463 611 661

2 9 2 18 74 22 16 13 15 31 9
3 14 4 61 80 58 31 59 55 113 68
4 15 8 62 85 131 99 69 134 175 81
6 22 16 63 111 140 126 102 158 342 147
8 24 32 103 120 185 167 129 225 344 220
9 25 39 114 167 187 256 152 247 351 418
12 40 45 121 180 217 273 169 337 392 457
13 59 64 125 270 231 290 300 356 418 612
16 62 67 139 274 256 333 334 362 429 634
18 64 78 188 293 337 393 348 425 489 658
[/CODE]

So, just for these 11 primes we need to do 10^10 CRM calculations to discover the lowest value. I have done one, using the first mod mentioned for each prime and I proud to announce that base b=51376801755005370229009845 produces the Sierpinski k-value with 11-cover of 38989969461035010680

The problem with this is that even after 10^10 calculations, there maybe other prime combinations that provide even smaller k. But it seems like a challenge worthy of this group, especially the programmers.

3. Requires that this k is the minimum. That would require exotic mathematics to find a really pesky base with no 2- 3- 4- 5- 6- 7- 8- 9- or 10-cover but 11-cover. I am confident that such base exists!! This is more for the mathematicians.

Of course a simpler idea might be to start with 7-cover.

1. The earliest base with 7-cover is b=2601 (only looking at primes less than 10^8.

2. smallest solution found for b<10000 is at b=4096 (such an interesting base!) with k=7183779570180

3. No progress

And 5-cover:

1. Lowest b is 339
2. Lowest k (checked to b=1000, for primes <10^8) is at b=339 and k= 84536206, but see 4096 below
3. No progress

Looking at b=4096 some astonishing facts:

7 new primes (p<10^8) at 5-cover, lowest k=327367
8 primes for 7-cover, lowest k=7183779570180
10 primes for 11-cover
11 primes at 13-cover
14 primes at 17 cover
10 primes at 19-cover
8 primes at 23 and 29-cover

Maybe someone can explain

Regards

Robert Smith

4096
 

4096 is a special base because it is a smooth power of 2.

Small (<10^8) primitive primes produced in the series (2^x)^11-1,x integer are as follows:

[CODE]x factors in x # primitives primitives
1 1^2 2 23,89
2 2 3 23,89,683
3 3 3 23,89,599479
4 2^2 5 23,89,397,683,2113
5 5 5 23,89,881,3191,201961
6 2*3 6 23,67,89,683,20857,599479
7 7 2 23,89
8 2^3 6 23,89,353,397,683,2113
9 3^2 5 23,89,199,153649,599479
10 2*5 7 23,89,683,881,2971,3191,201961
11 11 1 727
12 2^2*3 9 23,67,89,397,683,2113,20857,312709,599479,4327489
13 13 3 23,89,724153
14 2*7 5 23,89,617,683,78233
15 3*5 6 23,89,881,3191,201961,599479
16 2^4 7 23,89,353,397,683,2113,229153
17 17 2 23,89
18 2*3^2 3 23,89,683
19 19 3 23,89,599479
20 2^2*5 9 23,89,397,683,881,2113,2971,3191,201961
21 3*7 4 23,89,463,599479
22 2*11 2 727,117371
23 23 2 23,89
24 2^3*3 12 23,67,89,353,397,683,2113,7393,20857,312709,599479,4327489
25 5^2 5 23,89,881,3191,201961
26 2*13 5 23,89,683,2003,724153
27 3^3 6 23,89,199,153649,599479,8950393
28 2^2*7 8 23,89,397,617,683,2113,8317,78233
29 29 4 23,89,18503,64439
30 2*3*5 10 23,67,89,683,881,2971,3191,20857,201961,599479
[/CODE]

The table shows a deeper modular relationship, and where x shows a smooth factorisation, then it picks up primitive primes from factors of x. So x=24 (base 16777216) is particularly rich in primitive primes, not only for 11-cover, but the same applies to any p-cover, p prime.

This makes me think that I might be very wrong in assuming I will discover a 11-cover before b=200000. Since yesterday I have run up to b=171040 form 109794 with no b providing more that 8 primitives

Regards

Robert Smith

Errors and further comments on the first and second messages:

[QUOTE=robert44444uk;147084]
The second looks mind bending but an interesting exercise. The lowest k for a base might be provided by the 11 primes that are the lowest possible in b^11-1. Only certain primes are allowed and the lowest of these are:

p[1],p[2]....= 23,67,89,199,331,353,397,419,463,[B]611[/B],661
[/QUOTE]

611 is not prime, the table should have used 617, with the same mods

[QUOTE=robert44444uk;147084]
These each appear as factors of b^11-1 for 10 distinct mods of p[1],p[2]....= 23,67,89,199.... [/QUOTE]

i.e. these are the primes 1mod11

[QUOTE=robert44444uk;147084]
So, just for these 11 primes we need to do 10^10 CRM calculations to discover the lowest value. I have done one, using the first mod mentioned for each prime and I proud to announce that base b=51376801755005370229009845 produces the Sierpinski k-value with 11-cover of 38989969461035010680
[/QUOTE]

Because of the incorrect 611, the b created through CRM for the first line of mods becomes b= 77189447758048725094039008 and k= 2663654055214095820. Using the second line produces the smaller b=13112603617015536361200576 and k=613451437474032545. The quoted b still gave 11 primes, the last of which was 3917 in the place of 617

[QUOTE=robert44444uk;147084]
The problem with this is that even after 10^10 calculations, there maybe other prime combinations that provide even smaller k. But it seems like a challenge worthy of this group, especially the programmers.

[/QUOTE]

Finding minimum k by this approach looks to be a feasible exercise only if there are limited possibilities for minimum k. Can someone conclude that there are only two natural k values for a given series of primes, excluding cases with trivial cover?

Taking line 3 mods in the table provides b=44885449382568437090738444, and a lowest k of 613451437474032545, the same as b=13112603617015536361200576.

[QUOTE=robert44444uk;147084]
This makes me think that I might be very wrong in assuming I will discover a 11-cover before b=200000. Since yesterday I have run up to b=171040 form 109794 with no b providing more that 8 primitives[/QUOTE]

Investigated all bases with >8 1mod11 primes, (primes<10^8), to see if larger primes would also contribute and bingo! b=32400 has 11 !!! 23,89,331,2003,5743,14851,21121,37379,76649,103951,1750914287. The largest prime lies out the max for bigcovering.exe so it did not list it, and subsequently I cannot compute the k.

Should be easy to check that 32400 is the lowest base b with 11 1mod11 factors in b^11-1.

Posted a new sequence with OEIS

%I A146563
%S A146563 14,74,339,2601,32400
%N A146563 First instance prime-cover Sierpinski bases. Lowest base b such that k*b^n+1 can generate a Sierpinski number from cover sets with prime length. For example b=14 provides Sierpinski number k=4 such that 4*14^n+1 is always composite for any integer n. The covering set comprises 2 primes each providing prime factors for even or odd values of n in k*b^n+1, so called 2-cover, 2 = 1st prime. Series generated for 2-, 3-, 5- 7- and 11-cover
%H A146563 &lt;a href="http://www.mersenneforum.org/showthread.php?t=10872"&gt;Exotic Sierpinskis &lt;/a&gt;
%F A146563 To generate a member of the series, it is required to discover the lowest value of b such that b^p-1 has at least p prime factors of the form 1modp, excluding any p in b-1. The exclusion ensures that covers are not trivial, with all n being factored by a particular prime.
%e A146563 The corresponding k values providing the lowest Sierpinski numbers generated by known minimal k Sierpinski numbers for prime-covers are: 4*14^n+1 (2-cover) 2012*74^n+1 (3-cover) 84536206*339(n+1 (5-cover) unknown*2601^n+1 (7-cover) unknown*32400^n+1 (11-cover)
%o A146563 (Other) bigcovering.exe
%K A146563 hard,more,nonn
%O A146563 2,1
%A A146563 Robert Smith, Nov 01 2008

[QUOTE=robert44444uk;147084]I am trying to determine 3 things (in order of difficulty):

1. The lowest base with 11-cover
2. The lowest k for a base with 11-cover
3. The lowest base, if one exists that provides k with the minimum Sierpinski or Riesel k
[/QUOTE]

1. Recall that draft I sent you some time ago has the following line in it which answers your question.

[QUOTE]
The minimal base for longer prime period covers grows quickly: (p, minimal base b) = (2,14), (3,74), (5,339), (7,2601), (11, 32400), and (13,212574).
[/QUOTE]

2. For the lowest k, you should focus on k, not b. You need k*b^m+1 = 0 (mod p) where p divides b^11-1 (different p for each m in 0, 1, ..., 10), so then k^11 (b^11)^m = (-1)^11 (mod p), and it follows k^11 + 1 = 0 (mod p). So simply look for k^11 + 1 divisible by 11 primes and check whatever restrictions you are placing on k.