20100402, 09:55  #45 
May 2007
Kansas; USA
2×23×233 Posts 
I thought that everyone might find a list of GFN primes for bases <= 1024 up to n=131072 (2^17) somewhat interesting:
http://www.noprimeleftbehind.net/crus/GFNprimes.htm I now use the list for showing (or not showing) "k=x is a GFN with no known prime" on the web pages. Of interest is the "not possible" bases as opposed to the bases where a prime has not been found and is highly unlikely to ever be found. There are a total of 8 of them. Clearly there is a difference. If it's not possible, we can prove that there will never be a prime. If there is no prime shown, then it is possible because we cannot currently prove that there will never be one. Regardless, there is a very strong chance that no more primes will ever be found on these bases; at least with mathematics as we currently know it. The largest prime found was 150^2048+1. I realize that other projects have done such a search but I could not easily find a comprehensive list of them starting from n=1. Although they have all likely been contiguously searched to n=2^17, I wanted to doublecheck them all myself. It took about ~3 weeks on one core and I'm fairly certain that I was not using anywhere near the fastest method for sieving and testing. I suspect that other projects have likely pushed these bases to at least n=2^18 and possibly 2^19. Iirc, base 2 has been pushed to n=2^32 with no more primes found. I have added a reference link to this list at the bottom of the 1st post here because it gives some idea of why we exclude them from the conjectures. Gary Last fiddled with by gd_barnes on 20100402 at 10:03 
20100402, 11:57  #46  
Mar 2006
Germany
101110001000_{2} Posts 
Quote:
A paper of distribution of generalized Fermat prime numbers here (Downloadlink right hand). Last fiddled with by kar_bon on 20100402 at 12:09 

20100402, 16:33  #47  
May 2007
Kansas; USA
2×23×233 Posts 
Quote:
I see that there is one GFN prime on the top5000 for n=2^18 for base 24518. That tells me that likely someone has searched all bases up that high for n=2^18 so it appears that my search almost completely covered searches done for bases <= 1024. The one main exception is for bases <= 12, which have had largescale factorization done on them on the ProthSearch pages up to a much higher limit. It would be interesting to see something more uptodate on the search status since 2004. Thanks for the reference, Last fiddled with by gd_barnes on 20100402 at 16:40 

20100711, 23:18  #48 
Dec 2008
Boycotting the Soapbox
2^{4}×3^{2}×5 Posts 
Faster special case algorithm?
If we use Proth's theorem to search for primes of the form (k*2^n)^(2^m)+1 with k<2^n, and if (k*2^n)^2+m<2^m, we could do the required negacyclic convolution with SchoenhageStrassen in base k*2^n.
Choosing k<2^24 gives us 16M candidates for every (n,m) combination, involves much less arithmetic than irrational base DWTs and is suitable for GPUs. 
20100712, 00:32  #49  
May 2007
Kansas; USA
2·23·233 Posts 
Quote:
From what I could understand of it, since we search the forms k*b^n1 and k*b^n+1, it appears that the forms referred to here only allow for exponents (nvalues in our forms) that are powers of 2, which would only help in the search for GFNs, i.e. of the form b^(2^m)+1 where b=(k*2^n in your form). That wouldn't really help us much. To clarify my last para., it appears to me that in the form (k*2^n)^(2^m)+1, (k*2^n) is the base (the bvalue in our forms) and (2^m) is the exponent (the nvalue in our forms). 

20100712, 01:59  #50  
Dec 2008
Boycotting the Soapbox
2^{4}·3^{2}·5 Posts 
Quote:
Really? The idea would be to search for very large primes, for instance among the numbers that have 20*2^24 bits ~ 100Mdigits. With m=13 and k*2^n slightly less than 5*2^13 bits, we can do a negacyclic convolution of length 2^13 using 2^10 as a 2^13th root of 1 (mod 2^(5*2^14)+1). 

20100712, 02:55  #51  
May 2007
Kansas; USA
2×23×233 Posts 
Quote:
Edit: I see where the confusion comes from: A few posts ago where I posted all of the GFN primes for bases <= 1024 up to n=2^17. That was for my own reference as much as anything to see which bases <= 1024 did not have a GFN prime. It also helps our coadmin determine whether or not to put this statement on the web pages: "k=1 is a GFN with no known prime". But GFN primes are not needed to prove/disprove the conjectures. Therefore our list of GFN primes up to n=2^17 is sufficient for our purposes. You might post this in the GIMPS / math forums here. I think you'll get better response. I for one am interested in reading about faster ways to test GFNs for primes because for the # of candidates that need to be searched, they are very prime heavy. But I would do such a search outside of the scope of this project. Last fiddled with by gd_barnes on 20100712 at 03:07 

20100716, 05:31  #53 
May 2007
Kansas; USA
2·23·233 Posts 
I have added a new small subproject for CRUS. It is now reflected in the 1st post here as "subproject #2" and shown on the main web pages.
The subproject is to test the even k's for the Sierp base 2 2nd conjecture of k=271129. 3 projects have already extensively tested all odd k's < 271129. I have now tested all even k's. There are 3 k's remaining at n=100K. I will be testing them up to n=1M before releasing them for public sieving and testing. See the 1st post here for more info. All applicable info. including links for all of the projects involved in testing the Sierp 2nd conjecture is shown on the main powersof2 page and a new Sierp base 2 reservations page. I will be changing the "base 2 evenn and oddn" thread to include "base 2 evenk" and statuses will be reported there. I'm currently at n=100K on the k's and am doublechecking ALL remaining 77 k's for 78557<k<271129 at n=100K up to n=250K to make sure that nothing has fallen through the cracks of the 2 projects that are testing in that range except for the even k's. I should be at n=250K in ~2 weeks. After that, I'll just be testing the even k's so should go quickly. Interestingly, CRUS is already testing 2 Riesel base 2 even k's. If I find 1 of the 3 k's as prime by n=1M for Sierp base 2, we'll have 2 even k's on each side to search. Gary 
20100716, 06:27  #54 
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
2^{5}·11·17 Posts 
I don't understand. (k*2)*2^n+1=k*2^(n+1)+1 so all even ks have been tested as k/2. Do you mean for even ns as well or something?
An example I found of conversion: 55816*2^14536+1 = 6977*2^14539+1. 
20100716, 08:53  #55  
May 2007
Kansas; USA
10100111011110_{2} Posts 
Quote:
You could also do the same test on the 2 Riesel base 2 even k's that are remaining and have been tested to n>1M. Think: MOBs that we test all the time here. It has nothing to do with even n's. Bottom line: The base 2 conjectures that were originally proposed back in the 50s and 60s omitted a simple bit of mathematical reasoning in stating that the proof only needed primes for odd k's. They apparently assumed that even k's could always be "absorbed" into the exponent, which is not always the case. The Sierp base 4, Riesel/Sierp 5, and CRUS projects as well as Prof. Caldwell in his now published math paper on the Sierp conjectures have taken into account the testing of MOBs. This reasoning was confirmed and actually suggested/implied by Jean Penne early on in this project as well as Prof. Caldwell; both of whom have far more mathematical knowledge than me. Interestingly this did not come about until earlier this year when PrimeGrid decided to start fully "filling in" the Sierp base 2 2nd conjecture where PSP had only done prime k's and PrimeGrid was only doing the remaining odd k's. For the 1st conjecture of k=78557, there are no even k's remaining. Strangely, the highest first prime found for even k<78557 was, you guessed it, for k=55816 at n=14536. Was that just coincidence that you came up with k=55816 to use as an example? BTW reference your example, k=6977*2^3+1 is prime so k=6977 did not need searching further. But k=55816 could not claim such a small prime and so was not eliminated until n=14536. Last fiddled with by gd_barnes on 20100716 at 09:36 

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