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Generalized Fermat Prime Search :
Comprehensive GFN prime list for n=15 and above
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Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13620 ID: 53948 Credit: 266,787,098 RAC: 299,875

Iain has put together a very nice summary of all known GFN primes for n=15 and above:
http://www.primegrid.com/gfn_history.php
It includes all known GFN primes, not just those discovered at PrimeGrid.
Why only n=15 and above? Two reasons. Those are the ranges we're searching, and if you were to attempt to make a list of all known GFN primes at the smaller n's it would be an exceptionally long list. (There's almost 35 million known primes for n=2, for example.)
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My lucky number is 75898^{524288}+1  


I might add n=14 (and maybe n=13 eventually), as at least according to mackerel there might be some notsowelltested ranges to be found there.
 Iain
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3073428256125*2^12900001 is Prime!  

mackerelVolunteer tester
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Joined: 2 Oct 08 Posts: 2529 ID: 29980 Credit: 491,132,366 RAC: 1,515,223

My double check run on n=14 for b up to 2.7M is still in progress. Coincidently I checked on its progress earlier today. So far I have rediscovered 69 of the 89 known primes listed on Yves' page, and no new ones.
Just poked around the prime pages and was a bit surprised to see people have looked a lot higher at n=13 with the largest listed b at 113T! Makes 2.6M I tested to seem rather insignificant... I would be interested if it could be summarised who did what, and when.  

Yves GallotVolunteer developer Project scientist Send message
Joined: 19 Aug 12 Posts: 667 ID: 164101 Credit: 305,042,960 RAC: 0

A nice page... and a surprising result:
N = 32768, bMax = 10581514, expected primes = 124.12, found 147.
N = 65536, bMax = 3054284, expected primes = 37.69, found 45.
N = 131072, bMax = 1773312, expected primes = 11.2, found 16.
N = 262144, bMax = 1141182, expected primes = 4.4, found 7.
N = 524288, bMax = 847066, expected primes = 1.69, found 4.
Good news for the search but not for my conjecture.
The estimates are correct for N <= 16384 (196/196 in [2 ; 2.6M] at 8192 and 96/89 in [2 ; 2.7M] at 16384, thanks mackerel!).
But there are clearly more primes found than expected for N >= 32768.
Where can I find the number of candidates that were removed by the sieve? We can use these data to check the 'C_n'.
Yves
 

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13620 ID: 53948 Credit: 266,787,098 RAC: 299,875

A nice page... and a surprising result:
N = 32768, bMax = 10581514, expected primes = 124.12, found 147.
N = 65536, bMax = 3054284, expected primes = 37.69, found 45.
N = 131072, bMax = 1773312, expected primes = 11.2, found 16.
N = 262144, bMax = 1141182, expected primes = 4.4, found 7.
N = 524288, bMax = 847066, expected primes = 1.69, found 4.
More up to date information:
For n=32768, we have currently fully double checked up to b=13164678. According to http://yves.gallot.pagespersoorange.fr/primes/stat.html we should expect 152.21 primes. There are 168 known primes including 114 from PrimeGrid.
For n=65536, we have fully double checked through b=3596864. From your page, we should expect 43.86 primes and there are 25 primes from PrimeGrid and 27 nonPrimeGrid primes, for a total of 52. Edit: Numbers corrected as per axn's post below.
For n=1048576, we've full double checked to 721724 and have found 0 primes, with 0.81 expected. (Data set is likely too small to draw any conclusion.)
Where can I find the number of candidates that were removed by the sieve? We can use these data to check the 'C_n'
http://www.primegrid.com/sieving/gfn/
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My lucky number is 75898^{524288}+1  

Yves GallotVolunteer developer Project scientist Send message
Joined: 19 Aug 12 Posts: 667 ID: 164101 Credit: 305,042,960 RAC: 0

Where can I find the number of candidates that were removed by the sieve? We can use these data to check the 'C_n'
http://www.primegrid.com/sieving/gfn/
C_n and sieve data are in accordance.
We have C_n ~ e^gamma * log(p_max) * remaining / 100M,
where gamma is Euler's constant 0.577215665.
N = 32768, C_n = 5.8026588 ~ 5.805426
N = 65536, C_n = 11.195714 ~ 11.20249
N = 131072, C_n = 11.004301 ~ 11.01494
I have no explanation for the deviation...  

axnVolunteer developer Send message
Joined: 29 Dec 07 Posts: 285 ID: 16874 Credit: 28,027,106 RAC: 0

For n=32768, we have currently fully double checked up to b=13164678. According to http://yves.gallot.pagespersoorange.fr/primes/stat.html we should expect 152.21 primes. There are 168 known primes including 114 from PrimeGrid.
So, the gap has been narrowed to 16
For n=65536, we have fully double checked through b=3596864. From your page, we should expect 43.86 primes and there are 32 primes from PrimeGrid and 25 nonPrimeGrid primes, for a total of 57.
I count 52 primes below 3.6M @ T5K. That would mean that the gap has been holding steady.
If these observations are correct, it probably means that there is nothing really wrong with the estimation other than the vagaries of Poisson.
 

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13620 ID: 53948 Credit: 266,787,098 RAC: 299,875

For n=65536, we have fully double checked through b=3596864. From your page, we should expect 43.86 primes and there are 32 primes from PrimeGrid and 25 nonPrimeGrid primes, for a total of 57.
I count 52 primes below 3.6M @ T5K. That would mean that the gap has been holding steady.
Oops  you're correct. I erroneously counted all of PrimeGrid's primes rather just those below the current double check leading edge. There's only 25 primes found by PrimeGrid in that range, not 32. Added to the 25 nonPG primes and the total would be 50.
Our numbers are still off by two because there are two 65536 primes listed in PG's database which were actually discovered elsewhere long before PrimeGrid "found" them. Those were on neither my "PG" nor "nonPG" lists, so my number was low by two. The correct number is indeed 52.
I'll correct my post. Thanks.
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My lucky number is 75898^{524288}+1  

HonzaVolunteer moderator Volunteer tester Project scientist Send message
Joined: 15 Aug 05 Posts: 1902 ID: 352 Credit: 3,597,086,456 RAC: 5,375,587

On 21 Jan 2018 7:09:11, the 489,103 digit prime 29048042^65536+1 was found by Honza using the OCL2 (combined) app.
So, it becomes 274th PG GFN16 prime according to Primes by Project page.
Top5000 has 287 GFN16 primes. It's expected, some of them were found outside PG.
PG Prime by Project also has 857678^65536+1 and 843832^65536+1 found in early 2011.
Those are not reported on Top5000 even they would fit in back in 2011, hmm.
Is 843832^65536+1 really smallest GFN16 prime?
Nope, because GFN history page has those found by Yves during very early stages: 48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600.
Those are also not listed on Top5000.
GFN16 history page has 275 primes and should include 26+1 primes from Primes by Project page. This would make 302 GFN16 primes.
Would THIS be the most complete list?
Now, it would make a prime every ~96k b, with 200 lowest distance and 490062 highest distance.
Accordning to Yves page, we should expect 308 primes for <2<b<29055396.
This is in line with expectation :)
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My stats
Badge score: 1*1 + 5*1 + 8*3 + 9*11 + 10*1 + 11*1 + 12*3 = 186  


PG Prime by Project also has 857678^65536+1 and 843832^65536+1 found in early 2011.
Those are not reported on Top5000 even they would fit in back in 2011, hmm.
They have fallen off Top 5000 but are there: 857678^65536+1 and 843832^65536+1
The smallest GFN16 is known to be 48594^65536+1, attributed to Stephen Scott in June 2000.
EDIT: It is in fact possible to perform a search that gives what you seem to ask for. Typed 65536+1 in Description, typed Generalized Fermat in Comment, set radiobutton to "all verified primes", demanded 500 results, and asked for HTML. (Used a hack to take those choice out in the URL (HTTP GET instead of HTTP POST).)
/JeppeSN  

HonzaVolunteer moderator Volunteer tester Project scientist Send message
Joined: 15 Aug 05 Posts: 1902 ID: 352 Credit: 3,597,086,456 RAC: 5,375,587

They have fallen off Top 5000 but are there: 857678^65536+1 and 843832^65536+1
The smallest GFN16 is known to be 48594^65536+1, attributed to Stephen Scott in June 2000.
EDIT: It is in fact possible to perform a search that gives what you seem to ask for. Typed 65536+1 in Description, typed Generalized Fermat in Comment, set radiobutton to "all verified primes", demanded 500 results, and asked for HTML. (Used a hack to take those choice out in the URL (HTTP GET instead of HTTP POST).)
/JeppeSN
Yeah, I perfomed this search couple minutes ago when realized what I ahve ommited before...just before you posted.
And it listed 303 GFN16 primes.
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My stats
Badge score: 1*1 + 5*1 + 8*3 + 9*11 + 10*1 + 11*1 + 12*3 = 186  


The list of GFN n=16 is now uptodate and shows the same 303 primes.
 Iain
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Twitter: IainBethune
Proud member of team "Aggie The Pew". Go Aggie!
3073428256125*2^12900001 is Prime!  

Yves GallotVolunteer developer Project scientist Send message
Joined: 19 Aug 12 Posts: 667 ID: 164101 Credit: 305,042,960 RAC: 0

GFN15 passed the 100,000,000 milestone!
There are 1159 primes for b <= 10^8.
This is 13% more than expected (1020 primes). I think that this result has a statistical significance (it is unlikely to have occurred)... it should be investigated.
Congratulations to all the contributors!
 


GFN15 passed the 100,000,000 milestone!
There are 1159 primes for b <= 10^8.
This is 13% more than expected (1020 primes). I think that this result has a statistical significance (it is unlikely to have occurred)... it should be investigated.
Congratulations to all the contributors!
Interesting. Are you starting to doubt that your conjecture about the asymptotic occurrence of GFN primes is true?
Something else:
GFN20 has passed the b=1,000,000 milestone. I am looking forward to GFN20 (trailing edge) reaching b=1,048,576 at which point I will add one more term to OEIS A277967 (which I have "invented" myself).
/JeppeSN  

Yves GallotVolunteer developer Project scientist Send message
Joined: 19 Aug 12 Posts: 667 ID: 164101 Credit: 305,042,960 RAC: 0

Interesting. Are you starting to doubt that your conjecture about the asymptotic occurrence of GFN primes is true?
Yes but fortunately the error is a typo on the page http://www.primegrid.com/gfn_history.php#n15
As of 27th July 2018, there are 1114 known primes of the form b^2^15+1 = b^32768+1. is not correct, there are 1014 known primes.
Then there are 1059 primes for b <= 10^8 (3.8% more than expected).
Prob(count >= 1059) = 12%, this is not a significant difference :)  


Yes but fortunately the error is a typo on the page http://www.primegrid.com/gfn_history.php#n15
As of 27th July 2018, there are 1114 known primes of the form b^2^15+1 = b^32768+1. is not correct, there are 1014 known primes.
Sorry, will fix that shortly!
____________
Twitter: IainBethune
Proud member of team "Aggie The Pew". Go Aggie!
3073428256125*2^12900001 is Prime!  

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Comprehensive GFN prime list for n=15 and above 