Greetings All,

Following some discussions on the GFN1-12 Discord server about values of b for which there are likely no primes of the form b²ⁿ+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 b²ⁿ+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 CPU-days, the following term was found, and confirmed to be the lowest b for which b²ⁿ+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 so-called titanic primes (at least 1000 decimal digits).

Would be crazy if a longer chain of this sort could be found.

/JeppeSN

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?

dannyridel wrote:

"Fermat-ic" 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.

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 b²⁰. 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:

240164550712338756²⁰+1 (240164550712338756 + 1)
240164550712338756²¹+1 (240164550712338756^2 + 1)
240164550712338756²²+1 (240164550712338756^4 + 1)
240164550712338756²³+1 (240164550712338756^8 + 1)
240164550712338756²⁴+1 (240164550712338756^16 + 1)
240164550712338756²⁵+1 (240164550712338756^32 + 1)
240164550712338756²⁶+1 (240164550712338756^64 + 1)
240164550712338756²⁷+1 (240164550712338756^128 + 1)

This is a GFP-8, indicating a progression of length 8, despite the final term being b²⁷.

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 GFP-5, the first 148 b values which result in a GFP-6 and the first set of GFP-7 b values. Rob computed the remaining GFP-7 b values so that we have the first 101 known, and I finished off GFP-6 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:

GFP-5 (i = 0...4): https://oeis.org/A070694
GFP-6 (i = 0...5): https://oeis.org/A235390
GFP-7 (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 GFP-5 term #173 corresponds to the first GFP-6 and GFP-6 term #148 corresponds to the first GFP-7. We are working to determine the total number of GFP-7 to the first GFP-8, however the computation involved is significant and this will take some time.

The GFP-8 search also continues and should be completed to software limits (b=2⁶⁴) in the next few weeks. There should be ~5 more GFP-8 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 GFP-9 term is a substantial undertaking (estimate of 275,000 core-days, compared to the 75 it took to find the first GFP-8).

We will see where this takes us and report back with news and progress updates when things happen :)

Regards,
Kellen