To continue efforts to create complete list of small GFN primes up to b=2G, a "Search for Small Primes" project will be restarted March 02, 2022, 09:00 UTC on "Private GFN server".

Note this is not a PrimeGrid project. It's running on a separate server, server address is http://boincvm.proxyma.ru:30080/test4vm/ (also note the non-standard port). If you're a new user here, you must create your account on the site first, using invitation code PrimeGrid. Automatic registration from Boinc client most probably will not work.

Project description

The goal is to create full list of GFN primes as far as software limit and computation resources allows. Historically, each sequence is tested, depending on it's complexity, either up to b=400M (old software limit) either up to b=2G. Current testing status is:

GFN-1 - GFN-11 - tested manually up to b=2G by a group of PrimeGrid users;
GFN-12 - tested on Private GFN Server up to b=2G during first phase of this project;
GFN-13 - tested on Private GFN Server up to b=2G (400M-2G range tested during second phase of this project in 2021);
GFN-14 - tested on Private GFN Server up to b=400M (400M-2G range - in progress);
GFN-15 and above - currently tested on PrimeGrid.

In the third phase, we will test GFN-14 from b=400M to b=2G.

Primes found during this search will be relatively small and cannot be reported to any "top" list. We will find thousand of them. Although they'll be collected mostly for statistical purposes, these primes can be used as a starting points searching for "chains" or "twin" primes, which may lead to another interesting and reportable discoveries.

Technical details

The project is GPU only.

We'll use new generation of Yves Gallot' program, "Genefer 2.0". Designed exclusively to test small numbers (when each test takes just a few seconds), it can test many candidates at once, fully utilizing your GPU and requiring almost no CPU time. It also implements Gerbicz error checking to detect hardware errors in GPU calculations (unfortunately, fast double-checking is not possible yet).

Task size has been selected to be about 30 minutes on a 1660-class GPU. Since task runtime is proportional to log₂(b), first tasks will be completed faster. They'll reach expected length around b=300M and continue to grow slowly until end of project.

During first phase, when it was first time for Genefer 2.0 to be used in Boinc environment and on many different hosts, all tasks were fully double-checked. During second phase, a special reduced doublechecking scheme will be used. It requires only 25% of extra work (i.e. 1 special DC task will be sent for 4 normal tasks). This method differs from standard Boinc approaches, validation scheme will be difficult and non-standard. On site, you'll see that each workunit have only one result pending validation, which eventually will became validated (or not), and this delay can be very big.

Known issues

** No graceful error handling **

This program does not tolerates even slightest of GPU errors. If case of any calculation error, program will immediately abort with "Gerbicz failed" message. There are no restarts. Check GPU temperatures, underclock the GPU.

** A problem with multi-GPU systems **

A bug in version 1.00 causes program to always use first GPU in the system. It'll be fixed soon.
Update 1.01 fixed some multi-GPU issues, but it'll work only if your cards are of same type (i.e. all NVidia or all ATI). A combo of NVidia+ATI will not work. Correct fix is more complex. Hopefully it'll be done later.
Update 1.02 have combo issues fixed.

** Note **

Please avoid running this project on hosts which can interrupt GPU calculations often, especially on slow GPUs. For example, on systems which are configured to run Boinc or GPU tasks "only when user is not active", and user interrupts too often. Currently implemented checkpoint system saves only index of last fully processed candidate, but not a current state of each test. Although each individual test can be short (for example, 2 seconds), there are up to 256 test running in parallel. If a program will be interrupted again within 256*2 seconds, no progress can be recorded.

Published results

A list of all b-values can be downloaded from my disk https://disk.yandex.ru/d/BRQzejmjqpvVQ
This directory contains many GFN-related files.To get list of primes for GFN-12, look for "GFN12_primes_b_2G.7z" file. This file contains list of b-values, one per line. A file "readme_GFN0x.txt" contains short summary about status of all small GFN projects.

For GFN-13 and above, you may also take a look at PrimeGrid GFN History Page.

Interesting chains and sequences (GFN-12)

All primes were tested for possible sequences (Progressions, Twin primes and Cunningham chains). Results are posted below. Only newly discovered terms are shown here, it's assumed that corresponding b^n+1 is a GFN prime. Unfortunately, sequences made from GFN-12 primes are too small for T5K now.

Arithmetic progression, difference b^n-2 (3; b^n+1; 2*b^n-1)

2*418578174^4096-1
2*437703418^4096-1
2*515978626^4096-1
2*1666048606^4096-1
2*1701273208^4096-1
2*1730094120^4096-1
2*1829203026^4096-1
2*1946934480^4096-1

Cunningham chain of the second kind (p; 2p-1) - type 1 (b^n+1, 2*b^n+1)

2*330367536^4096+1
2*481426524^4096+1
2*517252674^4096+1
2*642693456^4096+1
2*809938152^4096+1
2*1117673412^4096+1
2*1201251558^4096+1
2*1379150574^4096+1

Cunningham chain of the second kind (p; 2p-1) - type 2 ((b^n)/2+1, b^n+1)

181770324^4096/2+1
1334346126^4096/2+1
1767327540^4096/2+1

Sophie Germain / start of Cunningham chain of the first kind (p; 2p+1)

2*90894850^4096+3
2*279872090^4096+3
2*1515334610^4096+3

Twin (p; p+2) (PRP)

1249625230^4096+3
1272384106^4096+3
1569303862^4096+3

Interesting chains and sequences (GFN-13 / 2000M)

Arithmetic progression, difference b^n-2 (3; b^n+1; 2*b^n-1)

2*398011242^8192-1
2*659807820^8192-1
2*967030564^8192-1

Cunningham chain of the second kind (p; 2p-1) - type 1 (b^n+1, 2*b^n+1)

2*103157148^8192+1 (T5K)
2*352666770^8192+1 (T5K)

Twin (p; p+2) (PRP)

188489906^8192+3
397931408^8192+3
716142476^8192+3
1785803204^8192+3

Interesting chains and sequences (GFN-14 / 400M)

Arithmetic progression, difference b^n-2 (3; b^n+1; 2*b^n-1)

2*307897022^16384-1

Twin entries are not proven primes, they're PRP due to their "+3" part, and their length is still too big to be proved by other methods.

Ready.....
____________
My lucky number 1059094^1048576+1

Go!!!
____________
My lucky number is 6219*2^3374198+1

Please avoid running this project on hosts which can interrupt GPU calculations often, especially on slow GPUs. For example, on systems which are configured to run Boinc or GPU tasks "only when user is not active", and user interrupts too often. Currently implemented checkpoint system saves only index of last fully processed candidate, but not a current state of each test. Although each individual test can be short (for example, 2 seconds), there are up to 256 test running in parallel. If a program will be interrupted again within 256*2 seconds, no progress can be recorded.

On a Vega 8 iGPU, there are 32 batches in total, with 64 tests per btach, I think the app detects the GPU and adjusts the batch size?

Hmm I've just noticed that there are 2048 tasks in total, in 8 batches which means 256 per batch, runtime on a GTX1650 is around 22 minutes.
On a Vega 8 iGPU, there are 32 batches in total, with 64 tests per btach, I think the app detects the GPU and adjusts the batch size? Runtime is around 5-6 hours.

Hmm I've just noticed that there are 2048 tasks in total, in 8 batches which means 256 per batch, runtime on a GTX1650 is around 22 minutes.
On a Vega 8 iGPU, there are 32 batches in total, with 64 tests per btach, I think the app detects the GPU and adjusts the batch size? Runtime is around 5-6 hours.

The number of tests per batch depends on the maximum work-group size. On Nvidia CL_KERNEL_WORK_GROUP_SIZE=1024 => 256 batches and on AMD CL_KERNEL_WORK_GROUP_SIZE=256 => 64 batches.
It is too bad this parameter is so small on AMD. This is a limitation of the driver, not of hardware. If this parameter was 1024, AMD GPUs would be more efficient for computations.

And @stream could we have a status page for this project like before? Would there be too many primes?

** A problem with multi-GPU systems **

A bug in version 1.00 causes program to always use first GPU in the system. It'll be fixed soon.

You could always forbid the second GPU to run that project in the app config file. Then at least you can run one GPU per machine on it.

** A problem with multi-GPU systems **
A bug in version 1.00 causes program to always use first GPU in the system. It'll be fixed soon.

When is "soon" roughly? Sometime during the challenge, or after?

Update 1.01 fixed most critical multi-GPU issues, but it'll work only if your cards are of same type (i.e. all NVidia or all ATI). A combo of NVidia+ATI will not work. Correct fix is more complex. Hopefully it'll be done later.

Update 1.01 fixed most critical multi-GPU issues, but it'll work only if your cards are of same type (i.e. all NVidia or all ATI). A combo of NVidia+ATI will not work. Correct fix is more complex. Hopefully it'll be done later.

Is it only my pc's not able to upload work? No problem getting new work and server status
seems ok, all is up and running, although browsing the site is slow.

A lot of users with heavy equipement, my guess db starting to fill up?

The project is 70% complete. With current (modest) participation level, testing of GFN-12 will be finished in 20-25 days, around April 01-03.

After completion of GFN-12, project will be paused until GFN-13 sieving reaches optimal level for new testing limits and application speed, then restarted on GFN-13, range 400M-2G (no estimated date yet).

Sorry, not sure where I should post this. The LLR2 testing subproject over at the private server shows a weird status, is this normal?

http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html

I'm currently seeing a lot of lines saying completed (highlighted in green), yet they have 0 in the "ready" and "in progress" columns as though no work was ever issued.

Also, can I tell from the task name on my computers which one I'm doing? For example "llr2_1225592_qqk9ud" or "llr2_1226200_qqke0i".

Where is this sieving taking place? Is it "PPS Sieve" in the main Primegrid? I want to help with the sieve.

The sieving for this project is manual sieving with gfsieve by Yves Gallot http://yves.gallot.pagesperso-orange.fr/gfsieve/. Up to 25,000P is complete in various intervals so far, but optimal for the project is ~75,000P. If anyone would like to help sieving please let me know either here or through private message and we would be more than happy for the help. The sieve software is a Windows executable and you can expect somewhere around 500P/day on an RTX 2070, or about 200P/day on a GTX 1080ti. Turing or Ampere are substantially more efficient with this sieving than other hardware (Volta would probably be good too, but there are few of us with any of those GPUs) and I am personally not running any instances on Pascal or older.

I suspended the GFN-13 sieving to help push GFN-12 forward and will resume once GFN-12 is completed. To do the sieving myself would take ~6-8 weeks past that point.

I would like to help. If you think my computers would be useful, let me know. I assume it only runs on GPUs? I've got 3 of Radeon R9 280X and 1 of Radeon RX 560. I've also got 9 varying CPUs from old quad cores up to a 24 core Ryzen 9 3900XT.

Yes, only on GPUs. And a reference speed for GTX1650 (better than a RX560) is 165P/day.

Will it run on AMD?

If it does, I've got the nice older ones (the 280X ones) with fast double precision if it uses that. Amazing on Milkyway for example.

And we're almost done with GFN-12! All candidates has been converted to workunits, and only 2800 workunits left unsent. When they'll be sent, project will be temporary paused until we finish additional sieving for GFN-13, then restarted on GFN-13. Stay tuned!

We're ready to continue!

Second phase of this project, GFN-13 search, will be started Monday, 26 April, at 09:00 UTC

In the second phase, we will test GFN-13 from b=400M (a point where previous GFN-13 search stopped) up to b=2G.

Notable changes comparing to GFN-12 search:

- The Genefer application has been tuned specially for GFN-13, speed improved by few percents.
- Issues with GPU combos (NVIDIA+ATI) should be fixed (although it wasn't tested on real setups yet).
- Since Genefer 2.0 has built-in Gerbicz check protecting against GPU errors, and it was found to be working very well during GFN-12 search, new reduced double-checking scheme will be used. An overhead of this scheme will be only 25% (i.e. 1 extra task per 4 "normal" tasks).
- Validation will be done completely "behind the scenes" and not visible on-site. You will see that workunit has only one result which is "pending validation". Eventually it'll become validated (or not). It happens because few independent workunits must be returned before decision can be made. In some cases, it'll take a while, so be patient. I'll monitor the situation to be sure that nothing is stuck.
- GFN-13 tests are about 4 times slower then GFN-12 (at least on my own GPU). So task size was reduced 4 times (512 candidates instead of 2048), and overall execution time should not change much.
- Credit coefficient was adjusted using benchmarks on my own GPU, to match PG GFN-17 more closely. I.e. you should get as much credit per hour as if you were running GFN-17 MEGA on PG. Of course every GPU is different, and your results may vary from my setup. But I expect that you'll receive more credit per hour, comparing to GFN-12.
- Individual prime lists will be available. But I'll need few days to write new statistics pages, so they'll be not available immediately on first day.

The project is not expected to be very long, just two times longer then GFN-12 (4x increase in test time compensated by reduced double-checking, also search range starts from 400M).

Good luck!

GFN-12 results are published, see top post

And... GFN-13 is already started and working just fine. Statistic and graphs are already available.

Great data. Looks like there are 201078 primes, which is 2.15 standard deviations below Yves's prediction of 202047.

- Issues with GPU combos (NVIDIA+ATI) should be fixed (although it wasn't tested on real setups yet).

With other projects (Einstein, Milkyway, Collatz), I've been running 2 or more tasks at once on the GPU to give extra throughput when waiting for CPU assistance. Should I be doing that on this project and on the main Primegrid?

GFN-13 should be 10-ish following that trend, although I'm sure someone with more maths in their brain has a calculated estimate. Yves Gallot wrote an excellent paper on this subject on August 4, 2020 titled "Gaps between generalized Fermat prime numbers", although I am not capable of calculating the expected number of consecutive primes as the math is well beyond my understanding. Perhaps he has a link to the paper and someone could use it to determine the expected. I only have my hard copy handy.

Top post updated, also added currently known results for GFN-13 and GFN-14 (both up to 400M).

25% done!

25% of this search is an equivalent of 400M range. An original GFN-13 search, which was first project on GFN server, tested candidates up to 400M, and it took more then 3 years. Now we did same range in just a 46 days!- thanks to new software, deeper sieving and, of course, your astounding participation!

There are no new special prime sequences (chains or progressions) yet.

Top post updated with 3 new GFN-13 special sequences, found up to 1000M:

2*659807820^8192-1
2*967030564^8192-1 (Arithmetic progressions)
716142476^8192+3 (Twin)

Alas, none of them are eligible for T5K.

The server is currently unreachable, this situation may stay longer then usual because it seems that I have to resolve some complex network / ISP issues. I'll try to deal with this situation as fast as possible.

50% done!

70% done!

Estimated date of completion shifted to 2022. Can we still finish this project in 2021? To achieve this, we must test 600K candidates per day. Our current rate stays stable around 400K candidates for a while.

I'm in progress of changing internet providers. Server is currently accessible from both uplinks / addresses, but DNS is already updated and full change to new connection should happen within next 24 hours. Please report any oddities like stuck uploads.

85% done!

Recent great boost in production shifted estimated completion date back to 2021, but there are two GPU challenges ahead. Can we still finish this project in 2021?

90% done, and we found new special sequence!

1785803204^8192+3 - Twin (p; p+2). This is a PRP so it cannot be reported to T5K.

(First post updated)

93.046 and climbing. :-)
Sure hope we finish it in December!!

95% done!

Only 10 days of work left at current rate, but there is a PrimeGrid GPU challenge ahead. Anyway, it seems that we can finish it in 2021!

It looks like the last set of tasks were generated within the last hour or so. Onto the home stretch now. Only ~6600 tasks left to go!

All GFN-13 tasks sent out

It's OK to join the PrimeGrid challenge now :)
(There are enough peoples to cleanup few aborted / timed out tasks when it becomes necessary).

GFN-13 project is finished!

All tasks returned and validated. I'll post another update after a day or two when testing of latest results will be finished - they must be verified by true primality test (Genefers tests are PRP) and tested for special sequences.

What a fast ride that was! Thanks! What would be the next step? Would it be increasing GFN-14 to b = 2G or at least higher to b > 400M? Or is that somewhat too demanding?

Or would it be taking lower GFN's to e.g. 5G?

Or a more controversial plan of taking GFN-15 from PG to PGS to 400M? (with the benefit that with Yves' new application GFN-15 can have a significant performance boost)

Or something else?

What a fast ride that was! Thanks! What would be the next step? Would it be increasing GFN-14 to b = 2G or at least higher to b > 400M? Or is that somewhat too demanding?

Or would it be taking lower GFN's to e.g. 5G?

Or a more controversial plan of taking GFN-15 from PG to PGS to 400M? (with the benefit that with Yves' new application GFN-15 can have a significant performance boost)

Or something else?

An obvious answer is "GFN-14", but it has some technical difficulties on the way. Yves' new program works best with small tests, where cost of initialization of the test becomes significant, comparing to small computation time of the test. As GFN-"N" grows, the difference between two programs decreases. New program checkpoints only when a full "vector" of tests is complete - which can be 128 or may be 256 tests long, it means that it may be tens of minutes between checkpoints or no checkpoints at all.

So GFN-15 is definitely no-no. GFN-14 may work, but may require some tweaking of the program. I didn't benchmarked yet, but it seems that GFN-14 task should have no more than 256 candidates to test, probably this number will be lowered to 192 or 128 (remember that there are almost no checkpoints, tasks should not be very long). Program should be able handle decreased sets of tests efficiently.

As for further expansion of the search up to 4G/5G/whatever, I don't see a reason to go so far. We collected enough primes and statistical data. Also, it means that all sieving must be re-done from scratch - all projects were sieved up to 2G. And I'm not sure that software (both Genefer and sieving programs) supports b >= 2^31.

If all software issues will be solved, I hope that we'll start new search in January 2022. Or, really, may be something else? I'm open for ideas.

Now a formal update about end of GFN-13 project which I've promised to write some time ago.

A search for GFN-13 primes was first public project run on my server. 400M range was finished in 3,5 years - it was a long way. Now we tested 400M-2000M range in 8 month!

First of all, such a great success was not possible without help all of you, who shared your resources with this project! Thank you very much! Also a major update of software, written by Yves Gallot, significantly improved performance on short types of tests and allowed "smart" verification scheme, reducing required amount of calculations.

The list of GFN-13 primes is now complete up to 2000M, without gaps, fully double-checked or smart-verified. Also we found one (alas, only one) new special sequence - twin PRP. These sequences seems to became rare beasts as "N" and "b" grows. See top post of this thread for published results and complete list of special sequences.

I'm still working on possible implementation of GFN-14 search - if everything goes well, we can start new search somewhere in the beginning of January, may be Jan 09. But it's still subject to change.

I'm still working on possible implementation of GFN-14 search - if everything goes well, we can start new search somewhere in the beginning of January, may be Jan 09. But it's still subject to change.

Anything new on the GN-14 search?

What is A259835(12)?