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Old 2016-11-23, 16:07   #1
sweety439
 
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Default 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^n-1, 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, b-1) = 1 (+ for Sierpinski, - for Riesel) and k*b^n+-1 (+ for Sierpinski, - for Riesel) is not prime for all n>=1"
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Old 2016-11-23, 20:29   #2
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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.
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Old 2016-11-24, 12:37   #3
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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 2016-11-24 at 13:01
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Old 2016-11-24, 15:47   #4
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I want the test limit for the GFNs, e.g. 22*22^n+1.
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Old 2016-11-24, 18:00   #5
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http://www.noprimeleftbehind.net/crus/GFN-primes.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 2016-11-24 at 18:15
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Old 2016-11-25, 09:26   #6
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Quote:
Originally Posted by gd_barnes View Post
http://www.noprimeleftbehind.net/crus/GFN-primes.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.
I guess you did, but I have to ask anyway...

Did you share your GFN search with Wilfrid Keller?

Luigi
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Old 2016-11-25, 11:35   #7
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Quote:
Originally Posted by ET_ View Post
I guess you did, but I have to ask anyway...

Did you share your GFN search with Wilfrid Keller?

Luigi
No. I am sure that he already has such trivial information. When I did my search the GFN project had already searched higher. I only ran it so that our project would have a list of small primes for bases <= 1030. It was done as more of a curiosity to see which bases had small GFN primes because CRUS does not consider GFNs in the testing of the bases.

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 2016-11-25 at 11:39
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Old 2016-11-28, 13:31   #8
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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)
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Old 2016-11-28, 13:39   #9
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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.
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Old 2016-11-28, 16:01   #10
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Quote:
Originally Posted by sweety439 View Post
The limits are actually higher than that. http://www.primegrid.com/stats_genefer.php
Code:
n=18    2027908    
n=19    1200598    
n=20    803136        
n=21    73132        
n=22    72590
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Old 2016-11-28, 16:51   #11
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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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