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Generalized Fermat Prime Search :
First Prime Pools (n=22, n=21 and n=20)
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Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13513 ID: 53948 Credit: 236,922,854 RAC: 3,199

I suppose we should start a pool to see who can guess which number will eventually be proven to be the new largest prime number. The form of this number will, of course, be b^4194304+1, with b ranging across all even numbers from b=2 to approximately b=475000. We know that all b<=2010 are composite, and testing is currently underway from 2012 through 5774. Most b values in the lower part of that range have already been proven composite, with just a few stragglers waiting for the double check to complete.
EDIT Nov 2012: Added a second pool for the first prime found in the new short GFN tasks at n=20.
The current upper and lower limits can always be checked on the GFN subproject status page.
For reference, here's a table of the lowest prime found at lower n values:
Min Min B as
n N Prime B % of N
1 2 2 100%
2 4 2 50%
3 8 2 25%
4 16 2 13%
5 32 30 94%
6 64 102 159%
7 128 120 94%
8 256 278 109%
9 512 46 9%
10 1,024 824 80%
11 2,048 150 7%
12 4,096 1,534 37%
13 8,192 30,406 371%
14 16,384 67,234 410%
15 32,768 70,906 216%
16 65,536 48,594 74%
17 131,072 62,722 48%
18 262,144 24,518 9%
19 524,288 75,898 14%
20 1,048,576 919,444 88% < n=20
21 2,097,152   < n=21
22 4,194,304   < n=22
23 8,388,608  
The first 2 columns are n and N, respectively. The third column is the lowest B such that B^N+1 is prime, i.e., the first prime found in that N. The fourth column shows the percentage of B to N  as you can see, there isn't a really strong correlation between min B and N.
Winner is the person who guesses closest to the actual B, regardless of whether they're above or below. Note that it's possible we may exhaust the search range without finding a prime, so "NONE" is an acceptable guess.
Just post to this thread to enter the pool.
You can amend your guess at any time by submitting a new guess.
I'll start the pool by guessing that the first prime will be found at B=75,898.
And, no, there's no prize for guessing correctly other than perhaps the nerdiest kind of fame. :)
Edit: Some people were concerned about picking numbers that have already been sieved out. Since the contest is for picking the number closest to the first prime, it really doesn't matter whether the number you pick is already known to be composite. In fact, there's no reason you can't pick an odd number if you wish. But for those who want to pick a number that isn't already proven composite, JimB provided these webpages listing the remaining values to be tested. If a number is in the appropriate list, it has NOT been removed by the sieve. In essence, to have a chance of not only being close, but picking the actual winning number, pick a number from these lists:
I don't expect JimB to be updating the lists often, or at all, but I also don't expect them to be changing rapidly.
____________
My lucky number is 75898^{524288}+1
 

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13513 ID: 53948 Credit: 236,922,854 RAC: 3,199

This post contains the current pool entries. It will be updated as new entries are received. To keep this thread clean, after moving the entry into this post, the user's posts will be deleted. (Similar to the way sieve reservation threads work.)
Current n=22 Pool Entries:
21,714 NeoMetal*
31,416 Gary Craig "I like PI."
41,944 [DPC]Pyrus
50,000 Scott Brown
58,312 KEP
65,536 Dave
68,000 DaveSun
72,000 Pooh Bear 27
73,618 Jeff17
75,898 Michael Goetz
81,984 JackTheSkipper
84,891 NeoMetal*
100,000 SteveRC
104,207 Roger
118,884 samuel7
144,000 serge
161,616 Hans Sveen
210,196 NeoMetal*
222,222 GDB
280,000 AXN
292,472 dannyridel
320,000 jsheridan
377,486 Uscapio Libertatis
510,510 Dirk Sellsted
2,238,721 Myrskylyhty
2,456,698 dukebg
5,194,572 Rafael
5,837,700 JeppeSN
NONE Konstantin Manz
NONE Christopher Siegert
Note regarding Gary Craig's imaginative "PI" entry: 3 and 31 are odd, and 314 and 3142 are already known to be composite. The next larger "PI" number would be 314,159, which is odd, and 3,141,592 is beyond the range of GeneferCUDA.
Current n=21 Pool Entries:
77,777 GDB
222,222 GDB
227,950 Robish
314,413 NeoMetal*
481,012 Hans Sveen
510,510 Dirk Sellsted
609,747 dannyridel
700,000 axn
777,777 GDB
1,122,018 Myrskylyhty
1,210,056 serge
1,265,314 dukebg
2,600,600 JeppeSN
5,194,572 Rafael
Current short n=20 Pool Entries:
41,751 NeoMetal*
75,898 Michael Goetz
82,514 Roger
116,496 Uscapio Libertatis
271,828 Gary Craig
290,710 Jeff17
333,333 GDB
677,772 gazzyk1ns
991,921 NeoMetal* *** WINNER *** (919444^2^20+1 is prime)
1,095,960 serge
1,200,000 axn
2,222,222 GDB
2,489,300 JeppeSN
5,194,572 Rafael
9,699,690 Dirk Sellsted
NONE Christopher Siegert
* Trailing edge is already past these guesses.
____________
My lucky number is 75898^{524288}+1  


With Van Zimmerman's n=20 find announced today, I guess we can declare NeoMetal* the winner of the n=20 pool. Good one.
We cannot see in the list of "pool entries" above at what date people have sent in their bet. My bet was much too high, but it was submitted quite recently when it was already established that n=20 had no "small" hits.
At what date was NeoMetal*'s winning bet submitted?
/JeppeSN  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13513 ID: 53948 Credit: 236,922,854 RAC: 3,199

Congratulations NeoMetal* for winning the n=20 first prime pool!
The winning entry of 991,921 was the closest to the actual prime's b value of 919,444.
____________
My lucky number is 75898^{524288}+1  


Resurrecting an old thread here.
Tried to look up the smallest GFNxx on the Prime Pages for xx between 10 and 20:
Smallest GFN10: 824^1024 + 1, 1986MMMdd, entrance rank 76
Smallest GFN11: 150^2048 + 1, 1986MMMdd, entrance rank 22
Smallest GFN12: 1534^4096 + 1, 1994Augdd, entrance rank 30
Smallest GFN13: 30406^8192 + 1, 1999Feb11, entrance rank 75
Smallest GFN14: 67234^16384 + 1, 1999Jun07, entrance rank 16
Smallest GFN15: 70906^32768 + 1, 2001May08, entrance rank 17
Smallest GFN16: 48594^65536 + 1, 2000Jun07, entrance rank 6
Smallest GFN17: 62722^131072 + 1, 2003Feb16, entrance rank 5
Smallest GFN18: 24518^262144 + 1, 2008Mar12, entrance rank 13
Smallest GFN19: 75898^524288 + 1, 2011Nov19, entrance rank 13, PrimeGrid
Smallest GFN20: 919444^1048576 + 1, 2017Sep02, entrance rank 12, PrimeGrid
Surprisingly, it looks like there was a period of about 11 months where a GFN16 was known, but no GFN15. But I think the interpretation is wrong. In any case, there is 167176^32768 + 1, 2000Mar05, entrance rank 8, so there was some GFN15 prime known (although not the smallest GFN15) before the smallest GFN16 was found.
Is it really true that the smallest GFN15 was missed initially, other GFN15s were found, then the smallest GFN16 was found, and only after that was the smallest GFN15 discovered? Maybe Yves remembers? EDIT: It seems correct. For example, on 2001Apr14 (web.archive.org), four GFN15 were known, but not the smallest one. On 2000Jun18, only one GFN15 (the second smallest) and one GFN16 (the smallest) were known.
/JeppeSN  

Yves GallotVolunteer developer Project scientist Send message
Joined: 19 Aug 12 Posts: 644 ID: 164101 Credit: 305,010,093 RAC: 78

Is it really true that the smallest GFN15 was missed initially, other GFN15s were found, then the smallest GFN16 was found, and only after that was the smallest GFN15 discovered? Maybe Yves remembers?
Yes, that's true.
GFN15, b in 210000 was tested in 1999. But then, because the running time didn't depend much on b, some "large" ranges were tested first. I was lucky and found 167176^{32768} + 1 in March 2000. Then we tried to find a largest GFN15 and found some. 70906^{32768} + 1 was found later in May 2001 when we filled in the gaps.
A single prime was found in [2; 100000] for GFN14 and the smallest known GFN15 was 167176^{32768} + 1 then a GFN16 in [2; 100000] was unexpected and GFN15 were some large primes in 2000 then many people searched for some GFN15.
I tested GFN16, b in 220000 and Steve Scott tested the range 20000100000 and succeeded.  


I feel the base 167'176 should be a Beagle Boy number, let me google: Yes, he has appeared. /JeppeSN  


I feel the base 167'176 should be a Beagle Boy number, let me google: Yes, he has appeared. /JeppeSN
LOL
____________
"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.  

GDBSend message
Joined: 15 Nov 11 Posts: 240 ID: 119185 Credit: 2,577,314,587 RAC: 18,994

How often are the pools updated? They haven't been updated since last August.
____________
 


How often are the pools updated? They haven't been updated since last August.
No one made a bet since I last did in August 2019
____________
SHSID Electronics Group
SHSIDElectronicsGroup@outlook.com
GFN14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1
 

GDBSend message
Joined: 15 Nov 11 Posts: 240 ID: 119185 Credit: 2,577,314,587 RAC: 18,994

How often are the pools updated? They haven't been updated since last August.
No one made a bet since I last did in August 2019
1 of the n=22, and 2 of n=21 entries have been surpassed. New entries might be posted if people saw that their guess was surpassed.
____________
 

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