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Greetings All,
Following some discussions on the GFN112 Discord server about values of b for which there are likely no primes of the form b^{2n}+1, we turned to the opposite case where a given b value may result in a long progression of prime values with increasing n.
The known minimum values for which all numbers of the form b^{2n}+1 are prime from n=0 to n=a were as follows, with no further terms know:
a(0): b=2
a(1): b=2
a(2): b=2
a(3): b=2
a(4): b=2
a(5): b=7072833120
a(6): b=2072005925466
Our resident GFN wizard, Yves, wrote a series of programs to efficiently sieve and test b values to search for the a(7) term and we are happy to announce that the search was successful! After ~ 75 CPUdays, the following term was found, and confirmed to be the lowest b for which b^{2n}+1 is prime for all n from 0 to 7.
a(7): b=240164550712338756
So, with that, we would love to find a solution for n=8, however we have reached a point where a small search is unlikely to find a solution for n=8, as that is a few orders of magnitude more difficult. Solutions for n=9 or n=10 are each that much more difficult again. However; these are not outside of the range of a distributed project, and the computational effort required for n=10 is comparable to that of the AP27 search, especially if a GPU implementation is created.
This is an exciting revival of an old search, and I hope that we are able to add more to the maximum progression length in the future!
Regards,
Kellen 



Really nice, and I see you are adding it to https://oeis.org/A090872.
Well done by Yves!
Here are the eight primes (not that I think the readers are idiots, just to celebrate them):
240164550712338756 + 1
240164550712338756^2 + 1
240164550712338756^4 + 1
240164550712338756^8 + 1
240164550712338756^16 + 1
240164550712338756^32 + 1
240164550712338756^64 + 1
240164550712338756^128 + 1
The last two are socalled titanic primes (at least 1000 decimal digits).
Would be crazy if a longer chain of this sort could be found.
/JeppeSN 



Really nice, and I see you are adding it to https://oeis.org/A090872.
Well done by Yves!
Here are the eight primes (not that I think the readers are idiots, just to celebrate them):
240164550712338756 + 1
240164550712338756^2 + 1
240164550712338756^4 + 1
240164550712338756^8 + 1
240164550712338756^16 + 1
240164550712338756^32 + 1
240164550712338756^64 + 1
240164550712338756^128 + 1
The last two are socalled titanic primes (at least 1000 decimal digits).
Would be crazy if a longer chain of this sort could be found.
/JeppeSN
"Fermatic" Progression of n primes :D lol
Also:
This is an exciting revival of an old search, and I hope that we are able to add more to the maximum progression length in the future!
Revival?
____________
SHSIDElectronicsGroup@outlook.com
waiting for a TdP prime...
Proth "SoB": 44243*2^440969+1




This is an exciting revival of an old search, and I hope that we are able to add more to the maximum progression length in the future!
Revival?
Yep! https://www.primepuzzles.net/puzzles/puzz_137.htm. Most recent term added May 10, 2007 after Yves discovered the first 6 :) 


Michael GoetzVolunteer moderator Project administrator
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dannyridel wrote: "Fermatic" Progression of n primes :D lol
I was thinking they could be called "Gallot Numbers" (or "Kellen Numbers"? I don't know the history of these), but Fermatic Progression is actually kind of good.
____________
My lucky number is 75898^{524288}+1 



dannyridel wrote: "Fermatic" Progression of n primes :D lol
I was thinking they could be called "Gallot Numbers" (or "Kellen Numbers"? I don't know the history of these), but Fermatic Progression is actually kind of good.
We have been calling them "GFP" for short, but if they were going to get a named series I vote for Gallot Numbers :)
Yves = Interested in GFP + Big Brain + All the hard work
Kellen = Interested in GFP + Lots of CPU cores ;) 


Yves GallotVolunteer developer Project scientist Send message
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GFP is great, because it can be Generalized or Gallot ;)
If we read Fermat's letter, this progression is in the same vein.
He built the sequence 2^{1}, 2^{2}, 2^{3}, ... and added one. He proved that they are composite if the exponent is not in the sequence (i.e. a power of two) and he thought that the other ones were primes because he didn't find their divisors.
Here the sequence is b^{1}, b^{2}, b^{3}, ... Similarly b^{n} + 1 is composite is n is not a power of two. We try to replace 3, 5, 17, 257 with a longer sequence of primes. 


robishVolunteer moderator Volunteer tester
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I agree, Gallot numbers, really interesting and incredibly cool. ๐
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 



Greetings Folks,
Lots of stuff going on with this search still, so it is probably a good time for an update.
The nomenclature has been sorted out a little more officially, as it can be a little confusing with the progressions starting from b^{20}. We have settled into "GFP" followed by a number indicating the length of the progression, rather than the maximum value of n in the final term of the progression.
To use the original find as an example:
240164550712338756^{20}+1 (240164550712338756 + 1)
240164550712338756^{21}+1 (240164550712338756^2 + 1)
240164550712338756^{22}+1 (240164550712338756^4 + 1)
240164550712338756^{23}+1 (240164550712338756^8 + 1)
240164550712338756^{24}+1 (240164550712338756^16 + 1)
240164550712338756^{25}+1 (240164550712338756^32 + 1)
240164550712338756^{26}+1 (240164550712338756^64 + 1)
240164550712338756^{27}+1 (240164550712338756^128 + 1)
This is a GFP8, indicating a progression of length 8, despite the final term being b^{27}.
With that sorted out; on to some real news.
Yves made specialized versions of his GFP programs to search for GFP of different lengths, and computed the first 1000 b values which result in a GFP5, the first 148 b values which result in a GFP6 and the first set of GFP7 b values. Rob computed the remaining GFP7 b values so that we have the first 101 known, and I finished off GFP6 to 1000 values.
These sequences, or updates to existing sequences, have now been submitted to OEIS by JeppeSN and are in various stages of approval. They can be found at the following links:
GFP5 (i = 0...4): https://oeis.org/A070694
GFP6 (i = 0...5): https://oeis.org/A235390
GFP7 (i = 0...6): https://oeis.org/A335805
The comments have not been approved yet, so they are not showing up, but the astute will notice that GFP5 term #173 corresponds to the first GFP6 and GFP6 term #148 corresponds to the first GFP7. We are working to determine the total number of GFP7 to the first GFP8, however the computation involved is significant and this will take some time.
The GFP8 search also continues and should be completed to software limits (b=2^{64}) in the next few weeks. There should be ~5 more GFP8 in this range, so there will be a new OEIS sequence shortly for those as well!
From here the search gets significantly more involved as sieve efficiency and search complexity, which have increased relatively uniformly, decouple. The search for the first GFP9 term is a substantial undertaking (estimate of 275,000 coredays, compared to the 75 it took to find the first GFP8).
We will see where this takes us and report back with news and progress updates when things happen :)
Regards,
Kellen 


robishVolunteer moderator Volunteer tester
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For those who are interested, the search continues :)
Progress so far .....
GFP8 progressions:
1
240164550712338756
3686834112771042790
6470860179642426900
7529068955648085700
10300630358100537120
16776829808789151280
17622040391833711780
19344979062504927000
More coming soon ;)
GFP9 May require assistance however, on a Boinc level. :)
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 


robishVolunteer moderator Volunteer tester
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Joined: 7 Jan 12 Posts: 1924 ID: 126266 Credit: 5,522,388,959 RAC: 2,954,432

Found a few more GFP8s
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 



Found a few more GFP8s
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
Beautifully done Rob! This is turning into quite the collection. The gap between the last two is pretty large too; hopefully that means that the next one is nearby :) 


robishVolunteer moderator Volunteer tester
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Cheers Kellen
Just for fun I wanted to examine the last one :)
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^1+1" d t4
46598149026149325546^1+1 is prime! (20 decimal digits, APRCL test)
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^2+1" d t4
46598149026149325546^2+1 is prime! (40 decimal digits, APRCL test)
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^4+1" d t4
46598149026149325546^4+1 is prime! (79 decimal digits, APRCL test)
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^8+1" d t4
46598149026149325546^8+1 is prime! (158 decimal digits).
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^16+1" d t4
46598149026149325546^16+1 is prime! (315 decimal digits)
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^32+1" d t4
46598149026149325546^32+1 is prime! (630 decimal digits)
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^64+1" d t4
46598149026149325546^64+1 is prime! (1259 decimal digits)
C:\Primes\LLR2\single>cllr64.exe q"46598149026149325546^128+1" d t4
46598149026149325546^128+1 is prime! (2518 decimal digits)
These are fun. Stay tuned for more. :)
The search for a GFP9 continues.......
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 


robishVolunteer moderator Volunteer tester
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Joined: 7 Jan 12 Posts: 1924 ID: 126266 Credit: 5,522,388,959 RAC: 2,954,432

Found another GFP8 ;)
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
15 51730769081270463240
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 



Found another GFP8 ;)
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
15 51730769081270463240
b is climbing fast!
____________
SHSIDElectronicsGroup@outlook.com
waiting for a TdP prime...
Proth "SoB": 44243*2^440969+1



robishVolunteer moderator Volunteer tester
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Joined: 7 Jan 12 Posts: 1924 ID: 126266 Credit: 5,522,388,959 RAC: 2,954,432

Found another GFP8 ;)
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
15 51730769081270463240
b is climbing fast!
350P/Day I think
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 


Yves GallotVolunteer developer Project scientist Send message
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b is climbing fast!
The expected number of GFPn in [2, B] is G_{n} * Integ[x = 2, B] dx/log(x)^{n},
where G_{8} = 3.6414e6 and G_{9} = 2.1045e7.
Then 22 GFP8 are expected in [2, 1e20].
The value for the expected smallest GFP9 is 8.7e21 which is about 893 GFP8, but the probability doesn't make sense for a single sequence. It just means that a GFP9 is a reasonable attempt.
The first GFP10 could be around 2,000,000 GFP8. 


robishVolunteer moderator Volunteer tester
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JeppeSN kindly updated here https://oeis.org/A337364
Thanks Jeppe :)
15 down, only 878 to go. Easy Peasy ;P
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 


robishVolunteer moderator Volunteer tester
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Joined: 7 Jan 12 Posts: 1924 ID: 126266 Credit: 5,522,388,959 RAC: 2,954,432

Found another GFP8 ;)
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
15 51730769081270463240
b is climbing fast!
I checked. It's 475P/day :)
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 


robishVolunteer moderator Volunteer tester
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Napkin Math. I'm getting on average one every 10 days, so it will take 24 years to get a GFP9 if it's at the far end. Hrmph :(
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 



Napkin Math. I'm getting on average one every 10 days, so it will take 24 years to get a GFP9 if it's at the far end. Hrmph :(
Likely the most important event of 2045. I see nothing listed on Wikipedia: 3rd millennium ยง 2040s. /JeppeSN 


robishVolunteer moderator Volunteer tester
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Found another GFP8
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
15 51730769081270463240
16 56592054138859807480
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 


robishVolunteer moderator Volunteer tester
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Joined: 7 Jan 12 Posts: 1924 ID: 126266 Credit: 5,522,388,959 RAC: 2,954,432

Found another GFP8
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
15 51730769081270463240
16 56592054138859807480
17 60908454029873490346
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 


robishVolunteer moderator Volunteer tester
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Joined: 7 Jan 12 Posts: 1924 ID: 126266 Credit: 5,522,388,959 RAC: 2,954,432

Another GFP8
1 1
2 240164550712338756
3 3686834112771042790
4 6470860179642426900
5 7529068955648085700
6 10300630358100537120
7 16776829808789151280
8 17622040391833711780
9 19344979062504927000
10 23949099004395080026
11 25348938242408650240
12 30262840543567048476
13 35628481193915651646
14 46598149026149325546
15 51730769081270463240
16 56592054138859807480
17 60908454029873490346
18 67486058769233442280
____________
My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26 



Another GFP8
Nicely done Rob! You're really cranking these out!
How far do you plan on going, or is this a "GFP9 or Bust" situation? You're coming up on 1% of the depth where 1 GFP9 is expected, so it isn't unrealistic to find one at the rate you're going.
Perhaps once the WW search has come to a close we could get this search here on PrimeGrid and go for a GFP10 or higher :)


