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Message boards : Generalized Fermat Prime Search : Comprehensive GFN prime list for n=15 and above

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Profile Michael GoetzProject donor
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The "Shut up already!" badge:  This loud mouth has mansplained on the forums over 10 thousand times!  Sheesh!!!Discovered the World's First GFN-19 prime!!!Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de PrimesFound 2 primes in the 2021 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (3,235,481)Cullen LLR Ruby: Earned 2,000,000 credits (3,624,591)ESP LLR Turquoise: Earned 5,000,000 credits (5,021,269)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,145,754)PPS LLR Sapphire: Earned 20,000,000 credits (20,751,038)PSP LLR Turquoise: Earned 5,000,000 credits (5,197,957)SoB LLR Sapphire: Earned 20,000,000 credits (36,067,618)SR5 LLR Jade: Earned 10,000,000 credits (10,007,110)SGS LLR Ruby: Earned 2,000,000 credits (3,718,606)TRP LLR Turquoise: Earned 5,000,000 credits (5,084,329)Woodall LLR Ruby: Earned 2,000,000 credits (2,963,361)321 Sieve (suspended) Jade: Earned 10,000,000 credits (10,061,196)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (22,885,121)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,035,522)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,051,121)AP 26/27 Jade: Earned 10,000,000 credits (10,118,303)WW Sapphire: Earned 20,000,000 credits (23,228,000)GFN Emerald: Earned 50,000,000 credits (77,916,132)PSA Jade: Earned 10,000,000 credits (12,445,029)
Message 90001 - Posted: 25 Nov 2015 | 16:29:10 UTC

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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321 LLR Gold: Earned 500,000 credits (597,175)Cullen LLR Amethyst: Earned 1,000,000 credits (1,355,381)ESP LLR Turquoise: Earned 5,000,000 credits (5,048,594)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,564,412)PPS LLR Amethyst: Earned 1,000,000 credits (1,025,115)PSP LLR Sapphire: Earned 20,000,000 credits (28,367,849)SoB LLR Jade: Earned 10,000,000 credits (16,215,329)SR5 LLR Turquoise: Earned 5,000,000 credits (5,189,992)SGS LLR Amethyst: Earned 1,000,000 credits (1,244,067)TRP LLR Turquoise: Earned 5,000,000 credits (5,169,405)Woodall LLR Amethyst: Earned 1,000,000 credits (1,070,956)321 Sieve (suspended) Bronze: Earned 10,000 credits (20,003)Cullen/Woodall Sieve (suspended) Silver: Earned 100,000 credits (200,371)Generalized Cullen/Woodall Sieve (suspended) Jade: Earned 10,000,000 credits (11,645,025)PPS Sieve Turquoise: Earned 5,000,000 credits (7,536,532)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,017,144)TRP Sieve (suspended) Gold: Earned 500,000 credits (554,946)AP 26/27 Ruby: Earned 2,000,000 credits (3,577,848)GFN Emerald: Earned 50,000,000 credits (97,047,976)PSA Ruby: Earned 2,000,000 credits (4,606,694)
Message 90002 - Posted: 25 Nov 2015 | 21:21:15 UTC

I might add n=14 (and maybe n=13 eventually), as at least according to mackerel there might be some not-so-well-tested ranges to be found there.

- Iain
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Discovered 6 mega primesEliminated 1 conjecture "k"Found 3 primes in the 2018 Tour de PrimesFound 1 mega prime in the 2018 Tour de PrimesFound 5 primes in the 2019 Tour de PrimesFound 6 primes in the 2020 Tour de PrimesFound 5 primes in the 2021 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (8,774,878)Cullen LLR Turquoise: Earned 5,000,000 credits (5,903,451)ESP LLR Turquoise: Earned 5,000,000 credits (6,454,573)Generalized Cullen/Woodall LLR Turquoise: Earned 5,000,000 credits (5,122,074)PPS LLR Double Bronze: Earned 100,000,000 credits (100,863,217)PSP LLR Jade: Earned 10,000,000 credits (15,223,714)SoB LLR Sapphire: Earned 20,000,000 credits (20,019,367)SR5 LLR Sapphire: Earned 20,000,000 credits (25,605,231)SGS LLR Turquoise: Earned 5,000,000 credits (7,360,621)TPS LLR (retired) Bronze: Earned 10,000 credits (34,130)TRP LLR Sapphire: Earned 20,000,000 credits (21,424,369)Woodall LLR Turquoise: Earned 5,000,000 credits (8,171,820)321 Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,236,219)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,383,853)Generalized Cullen/Woodall Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,626,419)PPS Sieve Emerald: Earned 50,000,000 credits (76,969,144)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Ruby: Earned 2,000,000 credits (2,293,882)TRP Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,012,757)AP 26/27 Sapphire: Earned 20,000,000 credits (21,918,894)WW Sapphire: Earned 20,000,000 credits (33,812,000)GFN Emerald: Earned 50,000,000 credits (76,982,057)PSA Ruby: Earned 2,000,000 credits (2,939,755)
Message 90003 - Posted: 25 Nov 2015 | 22:22:20 UTC - in response to Message 90002.

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.

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Message 90077 - Posted: 29 Nov 2015 | 18:37:48 UTC - in response to Message 90001.

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

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The "Shut up already!" badge:  This loud mouth has mansplained on the forums over 10 thousand times!  Sheesh!!!Discovered the World's First GFN-19 prime!!!Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de PrimesFound 2 primes in the 2021 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (3,235,481)Cullen LLR Ruby: Earned 2,000,000 credits (3,624,591)ESP LLR Turquoise: Earned 5,000,000 credits (5,021,269)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,145,754)PPS LLR Sapphire: Earned 20,000,000 credits (20,751,038)PSP LLR Turquoise: Earned 5,000,000 credits (5,197,957)SoB LLR Sapphire: Earned 20,000,000 credits (36,067,618)SR5 LLR Jade: Earned 10,000,000 credits (10,007,110)SGS LLR Ruby: Earned 2,000,000 credits (3,718,606)TRP LLR Turquoise: Earned 5,000,000 credits (5,084,329)Woodall LLR Ruby: Earned 2,000,000 credits (2,963,361)321 Sieve (suspended) Jade: Earned 10,000,000 credits (10,061,196)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (22,885,121)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,035,522)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,051,121)AP 26/27 Jade: Earned 10,000,000 credits (10,118,303)WW Sapphire: Earned 20,000,000 credits (23,228,000)GFN Emerald: Earned 50,000,000 credits (77,916,132)PSA Jade: Earned 10,000,000 credits (12,445,029)
Message 90078 - Posted: 29 Nov 2015 | 19:34:42 UTC - in response to Message 90077.
Last modified: 30 Nov 2015 | 6:33:32 UTC

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.pagesperso-orange.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 non-PrimeGrid 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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Message 90079 - Posted: 29 Nov 2015 | 23:42:02 UTC - in response to Message 90078.

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...

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321 LLR Gold: Earned 500,000 credits (656,177)Cullen LLR Silver: Earned 100,000 credits (457,297)ESP LLR Silver: Earned 100,000 credits (437,502)PPS LLR Amethyst: Earned 1,000,000 credits (1,256,334)SoB LLR Amethyst: Earned 1,000,000 credits (1,246,109)SR5 LLR Ruby: Earned 2,000,000 credits (2,193,129)SGS LLR Silver: Earned 100,000 credits (101,658)Woodall LLR Silver: Earned 100,000 credits (292,129)Cullen/Woodall Sieve (suspended) Bronze: Earned 10,000 credits (32,498)PPS Sieve Jade: Earned 10,000,000 credits (15,515,229)AP 26/27 Amethyst: Earned 1,000,000 credits (1,491,867)GFN Ruby: Earned 2,000,000 credits (4,245,870)PSA Silver: Earned 100,000 credits (101,309)
Message 90085 - Posted: 30 Nov 2015 | 6:03:52 UTC - in response to Message 90078.

For n=32768, we have currently fully double checked up to b=13164678. According to http://yves.gallot.pagesperso-orange.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 non-PrimeGrid 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.

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The "Shut up already!" badge:  This loud mouth has mansplained on the forums over 10 thousand times!  Sheesh!!!Discovered the World's First GFN-19 prime!!!Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de PrimesFound 2 primes in the 2021 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (3,235,481)Cullen LLR Ruby: Earned 2,000,000 credits (3,624,591)ESP LLR Turquoise: Earned 5,000,000 credits (5,021,269)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,145,754)PPS LLR Sapphire: Earned 20,000,000 credits (20,751,038)PSP LLR Turquoise: Earned 5,000,000 credits (5,197,957)SoB LLR Sapphire: Earned 20,000,000 credits (36,067,618)SR5 LLR Jade: Earned 10,000,000 credits (10,007,110)SGS LLR Ruby: Earned 2,000,000 credits (3,718,606)TRP LLR Turquoise: Earned 5,000,000 credits (5,084,329)Woodall LLR Ruby: Earned 2,000,000 credits (2,963,361)321 Sieve (suspended) Jade: Earned 10,000,000 credits (10,061,196)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (22,885,121)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,035,522)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,051,121)AP 26/27 Jade: Earned 10,000,000 credits (10,118,303)WW Sapphire: Earned 20,000,000 credits (23,228,000)GFN Emerald: Earned 50,000,000 credits (77,916,132)PSA Jade: Earned 10,000,000 credits (12,445,029)
Message 90086 - Posted: 30 Nov 2015 | 6:31:45 UTC - in response to Message 90085.

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 non-PrimeGrid 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 non-PG 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 "non-PG" lists, so my number was low by two. The correct number is indeed 52.

I'll correct my post. Thanks.
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Discovered 8 mega primesEliminated 4 conjecture "k"sFound 2 primes in the 2018 Tour de PrimesFound 1 prime in the 2018 Tour de Primes Mountain Stage2019 Tour de Primes largest primeFound 4 primes in the 2019 Tour de PrimesFound 1 mega prime in the 2019 Tour de PrimesFound 1 prime in the 2019 Tour de Primes Mountain StageFound 1 prime in the 2020 Tour de PrimesFound 4 primes in the 2021 Tour de PrimesFound 1 mega prime in the 2021 Tour de PrimesFound 1 prime in the 2021 Tour de Primes Mountain Stage321 LLR Double Bronze: Earned 100,000,000 credits (100,425,701)Cullen LLR Emerald: Earned 50,000,000 credits (96,556,711)ESP LLR Emerald: Earned 50,000,000 credits (50,853,190)Generalized Cullen/Woodall LLR Emerald: Earned 50,000,000 credits (50,309,119)PPS LLR Double Bronze: Earned 100,000,000 credits (155,424,135)PSP LLR Emerald: Earned 50,000,000 credits (50,563,867)SoB LLR Double Bronze: Earned 100,000,000 credits (113,220,736)SR5 LLR Double Bronze: Earned 100,000,000 credits (101,883,529)SGS LLR Emerald: Earned 50,000,000 credits (55,573,609)TPS LLR (retired) Bronze: Earned 10,000 credits (43,033)TRP LLR Double Bronze: Earned 100,000,000 credits (100,338,924)Woodall LLR Emerald: Earned 50,000,000 credits (50,956,094)321 Sieve (suspended) Double Bronze: Earned 100,000,000 credits (115,948,450)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,142,109)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,504,945)PPS Sieve Double Gold: Earned 500,000,000 credits (513,057,580)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,288,222)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,149,354)AP 26/27 Double Silver: Earned 200,000,000 credits (232,724,773)WW Double Silver: Earned 200,000,000 credits (401,748,000)GFN Double Gold: Earned 500,000,000 credits (777,188,770)PSA Double Gold: Earned 500,000,000 credits (535,232,795)
Message 113634 - Posted: 21 Jan 2018 | 13:28:26 UTC
Last modified: 21 Jan 2018 | 13:29:54 UTC

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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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (529,293)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (139,922)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Jade: Earned 10,000,000 credits (11,819,309)PSP LLR Silver: Earned 100,000 credits (212,242)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (145,419)SGS LLR Silver: Earned 100,000 credits (112,277)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)WW Turquoise: Earned 5,000,000 credits (9,628,000)GFN Amethyst: Earned 1,000,000 credits (1,707,013)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 113638 - Posted: 21 Jan 2018 | 17:41:59 UTC - in response to Message 113634.
Last modified: 21 Jan 2018 | 18:04:16 UTC

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

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Discovered 8 mega primesEliminated 4 conjecture "k"sFound 2 primes in the 2018 Tour de PrimesFound 1 prime in the 2018 Tour de Primes Mountain Stage2019 Tour de Primes largest primeFound 4 primes in the 2019 Tour de PrimesFound 1 mega prime in the 2019 Tour de PrimesFound 1 prime in the 2019 Tour de Primes Mountain StageFound 1 prime in the 2020 Tour de PrimesFound 4 primes in the 2021 Tour de PrimesFound 1 mega prime in the 2021 Tour de PrimesFound 1 prime in the 2021 Tour de Primes Mountain Stage321 LLR Double Bronze: Earned 100,000,000 credits (100,425,701)Cullen LLR Emerald: Earned 50,000,000 credits (96,556,711)ESP LLR Emerald: Earned 50,000,000 credits (50,853,190)Generalized Cullen/Woodall LLR Emerald: Earned 50,000,000 credits (50,309,119)PPS LLR Double Bronze: Earned 100,000,000 credits (155,424,135)PSP LLR Emerald: Earned 50,000,000 credits (50,563,867)SoB LLR Double Bronze: Earned 100,000,000 credits (113,220,736)SR5 LLR Double Bronze: Earned 100,000,000 credits (101,883,529)SGS LLR Emerald: Earned 50,000,000 credits (55,573,609)TPS LLR (retired) Bronze: Earned 10,000 credits (43,033)TRP LLR Double Bronze: Earned 100,000,000 credits (100,338,924)Woodall LLR Emerald: Earned 50,000,000 credits (50,956,094)321 Sieve (suspended) Double Bronze: Earned 100,000,000 credits (115,948,450)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,142,109)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,504,945)PPS Sieve Double Gold: Earned 500,000,000 credits (513,057,580)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,288,222)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,149,354)AP 26/27 Double Silver: Earned 200,000,000 credits (232,724,773)WW Double Silver: Earned 200,000,000 credits (401,748,000)GFN Double Gold: Earned 500,000,000 credits (777,188,770)PSA Double Gold: Earned 500,000,000 credits (535,232,795)
Message 113639 - Posted: 21 Jan 2018 | 18:07:58 UTC - in response to Message 113638.
Last modified: 21 Jan 2018 | 18:08:41 UTC

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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Message 113659 - Posted: 22 Jan 2018 | 10:53:23 UTC - in response to Message 113639.

The list of GFN n=16 is now up-to-date and shows the same 303 primes.

- Iain
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Message 120865 - Posted: 2 Oct 2018 | 13:59:14 UTC

GFN-15 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!

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Message 120867 - Posted: 2 Oct 2018 | 15:53:36 UTC - in response to Message 120865.

GFN-15 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:

GFN-20 has passed the b=1,000,000 milestone. I am looking forward to GFN-20 (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

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Message 120886 - Posted: 3 Oct 2018 | 9:55:18 UTC - in response to Message 120867.

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 :-)

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Message 120889 - Posted: 3 Oct 2018 | 11:42:18 UTC - in response to Message 120886.


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!
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