20161128, 17:10  #12 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
111001100010_{2} Posts 
https://www.utm.edu/staff/caldwell/preprints/2to100.pdf
There is another researcher for the Sierpinski numbers base b (for 2<=b<=100), but he or she thinks the k's with full or partial algebraic factors (e.g. 2500*16^n+1) as Sierpinski number, but excluding the GFNs in the research. (The list in the final page has many errors, e.g. k=4 is remaining in the Sierpinski base 53 problem, but the list only lists {1816, 1838, 1862, 1892}) 
20161129, 07:39  #13  
"Gary"
May 2007
Overland Park, KS
2×19×317 Posts 
Quote:
http://www.noprimeleftbehind.net/cru...espowers2.htm If we were to allow partial or algebraic factors to become the conjecture of each base on this page, bases 2 evenn, 4, 16, 64, 256, and 1024 would have a conjecture of k=9 and all would be quickly proven. The powerof2 bases are some of our most interesting and to have nearly half of them quickly proven would be mathematically uninteresting. Also if you look at the main Riesel conjectures page you will see a relatively high percentage of bases that have algebraic exclusions. Many of these conjectures, especially among the lowest bases, would become uninteresting. A second reason that we chose to do this is that we had an individual early in our project who created a program called covering.exe. The program was very effective and efficient at coming up with the lowest k for each base that contained a full "numeric" covering set. To answer a question from another post that you had: The base 2 evenn and oddn searches are technically a completely different effort and are referred to as the LiskovetsGallot conjectures. See here: http://www.primepuzzles.net/problems/prob_036.htm CRUS chose to include them in our project not long after our project started for three reasons: (1) They were interesting and appeared provable in our lifetimes. [1 of the 4 bases was proven just in the last year with some huge primes and the other three have <= 4 k's remaining.] (2) They were relatively similar to our project and involve our lowest base 2, which is the easiest for us to search. (3) We could combine the sieving effort with base 4 and other powersof2 bases. The conjectures are not the same as base 4. They only include k where k==3 mod 6 and the trivial k's are different. There are many overlapping primes and k's remaining but if you look closely at our pages you will see how different that they are. Last fiddled with by gd_barnes on 20161129 at 08:01 

20161129, 07:53  #14  
"Gary"
May 2007
Overland Park, KS
2·19·317 Posts 
Quote:
In doing this research I think you can see by how high the lowest base with a GFN prime is as a reason why GFNs must be excluded from our searches. Although it cannot be proven, there are unlikely to be any more GFN primes for b<=1030. Even if one did somehow pop up sometime in the distant future for one of our over 500 bases, for each specific base the chances are extremely small and mathematicians believe that the number of GFN primes for each base is finite. Last fiddled with by gd_barnes on 20161129 at 08:03 

20161129, 13:56  #15 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
However, there may be infinitely many primes of the form 1*2^n+1, just as that there may be infinitely many primes of the form 3*2^n+1. I think the number of both type of primes are infinite. For example, 2^(2^34)+1 may be prime.
Last fiddled with by sweety439 on 20161129 at 14:02 
20161129, 18:46  #16 
"Gary"
May 2007
Overland Park, KS
2×19×317 Posts 
Consider doing more research before making such statements.

20161206, 20:52  #17 
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
1011110101001_{2} Posts 
Should CRUS consider the situation where there is a full covering set for a k even if odd or even ns are also eliminated with a partial algebraic factorization? As long as there is a full covering set surely the partial algebraic factorization shouldn't matter when selecting a Reisel/Sierpinski k.

20161207, 06:47  #18  
"Gary"
May 2007
Overland Park, KS
2×19×317 Posts 
Quote:
Last fiddled with by gd_barnes on 20161207 at 06:48 

20161207, 08:15  #19 
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
3^{2}×673 Posts 
It is possible to have a full fixed numeric covering set for a k as well as having having algebraic factors. I suppose the algebraic factors mean that it is more likely that there is a covering set due to two composites having potential factors.

20161208, 03:24  #20 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
23·439 Posts 
The covering factors of the algebraic composites will "leak through" into the "normal" covering set.
This happens in the CRUS set frequently. Take the metaseries R14, R44, R74, R104, etc  R(30q+14)  they all have CK=4. So, you could say that for each of them, even n's are eliminated algebraically, or you can say that they are eliminated by factor 3; and that odd n's are eliminated by factor 5. Potaetoe, potahtoh. 
20161210, 13:01  #21  
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
3^{2}·673 Posts 
Quote:
ns in 4*14^(2n1)1 are divisible by 5. It feels a little artificial to me for the covering set for the even ns be based upon the covering sets for the two algebraic factors overlapping like that. It would be much nicer if there was a full covering set for all the three forms above. How many bases would have their CK extended if that logic was used? 

20161215, 19:45  #22 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
Another question:
Why you does not consider 8 and 32 as Sierpinski numbers base 128? All numbers of the form 8*128^n+1 and 32*128^n+1 are composite, and they have no algebra factors. Last fiddled with by sweety439 on 20161215 at 19:46 
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