20161123, 16:07  #1 
Nov 2016
2^{3}·5·59 Posts 
Definition of Sierpinski/Riesel number base b
Hi,
I am a newcomer, I understand your definition of Sierpinski/Riesel number base b. However, I want do you know why you don't think a number k with k*b^n+1 composite for all n>=1, but with all or partial algebraic factors (e.g. 8*27^n+1, 2500*16^n+1, 9*4^n1, etc.) as Sierpinski/Riesel number? Besides, I think the GFNs (e.g. 22*22^n+1) can also be in the conjecture, since nobody knows whether there exists an n such that 22*22^n+1 is prime, just as that nobody knows whether there exists an n such that 5128*22^n+1 is prime. I think the definition of Sierpinski/Riesel number base b should be "a positive integer k such that gcd(k+1, b1) = 1 (+ for Sierpinski,  for Riesel) and k*b^n+1 (+ for Sierpinski,  for Riesel) is not prime for all n>=1" 
20161123, 20:29  #2 
May 2007
Kansas; USA
2·5,113 Posts 
The main reason: k's with algebraic factors do not have a single set of fixed numeric factors. From our perspective the conjectured k must have a fixed set of numeric factors.
Second: Many of the conjectures would become "not interesting" (mostly on the Riesel side) if a k with partial or algebraic factors to make a full covering set were allowed to become the conjecture. Many would have a small conjecture and quickly be proven. It is relatively simple to identify such k's and eliminate them from testing just like we do with k's that have tri Third: Software was created early in the project that quickly and accurately identifies the lowest conjectured k with a known covering set of numeric factors. 
20161124, 12:37  #3 
Nov 2016
100100111000_{2} Posts 
How about the GFNs? e.g. 22*22^n+1, since nobody knows whether there exists an n such that 22*22^n+1 is prime, just as that nobody knows whether there exists an n such that 5128*22^n+1 is prime. The k=22 can be in the Sierpinski base 22 conjecture, just as k=5128. For the trivial k's, e.g. 34*22^n+1, is always divisible by 7. Thus, all numbers of the form 34*22^n+1 are composite and k=34 cannot be in the Sierpinski base 22 conjecture. However, nobody knows whether all numbers of the form 22*22^n+1 and 5128*22^n+1 are composite, so k=22 and k=5128 should be in the Sierpinski base 22 conjecture. (The first conjectured base 22 Sierpinski number is still 6694)
Last fiddled with by sweety439 on 20161124 at 13:01 
20161124, 15:47  #4 
Nov 2016
2^{3}×5×59 Posts 
I want the test limit for the GFNs, e.g. 22*22^n+1.

20161124, 18:00  #5 
May 2007
Kansas; USA
2×5,113 Posts 
http://www.noprimeleftbehind.net/crus/GFNprimes.htm
All GFNs have been searched to n>=2^17 personally by me. But I am clearly not the only one interested in such an effort. There is a GFN project out there that has likely searched them to n=2^19 or maybe n=2^20. It is highly unlikely that any more GFN primes will be found in the foreseeable future for b<=1030. Note that 22*22^n+1 is the same as 22^(n+1)+1 so the search depth for GFNs where k<>1 can be extrapolated from those pages. GFNs are excluded from the project and the conjectures because only n=2^m where m>=0 can be prime. Mathematicians have agreed that the number of primes of such forms are finite. Therefore it cannot be known if such forms will contain a prime. In other words 22*22^n+1 is very different from 5128*22^n+1. Last fiddled with by gd_barnes on 20161124 at 18:15 
20161125, 09:26  #6  
Banned
"Luigi"
Aug 2002
Team Italia
2^{5}·149 Posts 
Quote:
Did you share your GFN search with Wilfrid Keller? Luigi 

20161125, 11:35  #7  
May 2007
Kansas; USA
10226_{10} Posts 
Quote:
The highest prime that I found was 150^(2^11)+1 and all bases <=1030 were searched to n=2^17. With today's software and machines it would be extremely trivial to doublecheck and recreate the list. Last fiddled with by gd_barnes on 20161125 at 11:39 

20161128, 13:31  #8 
Nov 2016
2^{3}·5·59 Posts 
Why the CRUS includes the "base 2 even n" and the "base 2 odd n" conjectures, I think they are the same as the base 4 conjectures. (they are equivalent to the base 4 conjectures when k = 0 (mod 3). Besides, the k != 0 (mod 3) in the base 4 conjectures are equlivalent to the base 2 conjectures)

20161128, 13:39  #9 
Nov 2016
2^{3}·5·59 Posts 
All of the GFNs with base b<=1030 (see http://oeis.org/A243959)) are searched to n=2^19, no primes found for n>2^11.
Since the smallest n such that n^(2^19)+1 is prime is 75898, n^(2^19)+1 is composite for all 2<=n<=1030. Besides, according to http://oeis.org/A244150, the smallest n such that n^(2^18)+1 is prime is 24518, n^(2^18)+1 is also composite for all 2<=n<=1030. For n^(2^20)+1, since 75898 > 275^2, n^(2^20)+1 is composite for all 2<=n<=275. 
20161128, 16:01  #10  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,281 Posts 
Quote:
Code:
n=18 2027908 n=19 1200598 n=20 803136 n=21 73132 n=22 72590 

20161128, 16:51  #11 
Nov 2016
2^{3}×5×59 Posts 
72590 > 269^2. Thus, n^(2^23)+1 is composite for all 2<=n<=269.
269 > 16^2. Thus, n^(2^24)+1 is composite for all 2<=n<=16. Therefore, the test limit for the GFNs are: b=2: 2^32 (https://web.archive.org/web/20151125...et/fermat.html) b=4: 2^31 (the same as b=2) b=6: 2^27 (https://web.archive.org/web/20151122...net/GFN06.html) b=8: algebra factorization b=10: 2^23 (https://web.archive.org/web/20151122...net/GFN10.html), but now 2^24 b=12: 2^23 (https://web.archive.org/web/20151122...net/GFN12.html), but now 2^24 b=14: 2^24 b=16: 2^30 (the same as b=2) 18<=b<=268: 2^23 270<=b<=72588: 2^22 
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