Exotic covers

[robert44444uk]

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

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

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

[robert44444uk]

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:

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

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

[robert44444uk]

Errors and further comments on the first and second messages:

Originally Posted by robert44444uk

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

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

Originally Posted by robert44444uk

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

i.e. these are the primes 1mod11

Originally Posted by robert44444uk

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

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

Originally Posted by robert44444uk

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.

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.

Originally Posted by robert44444uk

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

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.

[robert44444uk]

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

[PrimeMogul]

Originally Posted by robert44444uk

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

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

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).

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.