Welcome to the AP27 Search (Arithmetic Progression of 27 primes)

An arithmetic progression is a sequence of numbers with a common difference between any two successive numbers in the sequence. For instance, the sequence 3, 5, 7, 9, 11, 13, 15, ... is an arithmetic progression with a common difference of 2.

Therefore, an arithmetic progression of primes is a sequence of primes with a common difference between any two successive numbers in the sequence. For example 3, 7, 11 is an arithmetic progression of 3 primes with a common difference of 4.

For an arithmetic progression (AP) of primes, AP-k is k primes of the form p + d*n for some d (the common difference between the primes) and k consecutive values of n. The above AP-3 is 3 + 4*n for n=0,1,2.

n=0; 3 + 4*0 = 3 + 0 = 3
n=1; 3 + 4*1 = 3 + 4 = 7
n=2; 3 + 4*2 = 3 + 8 = 11

Another example is the AP-10 of the form 199 + 210*n for n=0..9. This produces the following sequence: 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089.

AP-k is also sometimes notated PAP-k (Primes in Arithmetic Progression).

We are searching for the longest AP, not the largest. Currently the longest known AP is of length 26. The first known AP26 was discovered on April 12th, 2010 here at PrimeGrid by Benoãt Perichon ([AF>HFR>RR] Jim PROFIT) of France. Since then, three additional AP26s have been found, one by James Fry, and two of them by Bryan Little (mfl0p). For a complete list of records including longest and largest AP's, please see Jens Kruse Andersen's Primes in Arithmetic Progression Records. Also, a list of the top 20 largest (not longest) AP's can be found at The Prime Pages: The Top 20.

New AP discoveries will be announced in the New APs forum thread.

Jaroslaw Wróblewski's AP26 algorithm will be used for the search. The AP26 algorithm is capable of finding AP27 and AP28 progressions as well as the shorter progressions.

The original AP26 program was written by Jaroslaw Wróblewski and adapted to BOINC by Geoff Reynolds.
The 2016 CPU and OpenCL versions of AP26 were updated by Bryan Little and Iain Bethune.

All builds by Brian Little and Iain Bethune

Additional information can be found here:

How to search for 26 primes in arithmetic progression? by Jaroslaw Wróblewski
Primes in arithmetic progression - Wikipedia
Prime Arithmetic Progression - Wolfram MathWorld
arithmetic sequence - The Prime Glossary at the Prime Pages

How to Participate?
Go to your PrimeGrid preferences page and select AP27.

Credit is currently set at 4043 cobblestones per WU.

WU expectations
Current WU's test 100 K's at 10 shifts each. The 2008-2010 AP26 tasks, at the end, tested 9 K's at 1 shift, so these tasks are about 111 times larger.

If K is divisible by one of the primes in the range 29-59, this K is ignored, i.e. AP26 quits instantly. Those are just over 18% of all K's.

If K is divisible by 61, the time is increased by some 20% over an "ordinary" K and if a few next primes (67, 71, ...) divide K it can be even longer.

Progression of WU's
We are initially sending out work at both shift=0 and shift=640. No decision has been made about when to start searching additional shifts.

The tasks with shift=0 overlap somewhat with the original AP26 search and serves as a double check in case anything was missed. The original AP26 project searched six shifts to the following levels. Remember that each AP26 shift was 64 bits wide, whereas an AP27 shift is 640 bits wide. Our current shift therefore covered all six of the original AP26 shifts.

AP26 (2008) progress:
shift=0; k<43902416
shift=64; k<33973721
shift=128; k<25165464
shift=192; k<16288724
shift=256; k<9354514
shift=320; k<2751343

Initial AP27 progress: (24 August 2016 to 20 Sep 2016)
shift=0; k<2538946
shift=640; k<2538946

For the app name do we use ap26 or ap27?
It seems that ap26 is still active. Thanks

Also the link on the Preferences Page takes you to AP26 not This Thread.

No Biggie on the app name ap26 works fine.
One more question on this Project, as far as the GTX Titan Graphics Cards would enabling Double Precision floating point that this card has?
I know very little BOINC Projects can use this setting and even slows down GPU Projects other then MilkyWay@home that is.
Thanks

Have you already tried a shortcut for possibly finding AP27s - use the same starting points as all known AP26s, to see if any of them are actually the first part of an AP27?

So this is still the same app as for the AP26 search, CPU and GPU-wise?

So this is still the same app as for the AP26 search, CPU and GPU-wise?

Won't get work for AP26 ocl

Not enough memory on the card. You need 1.5G, and it won't fetch tasks if BOINC detects less than that.

What have i missed?

Hm, even in standalone-mode via the console i do need 1.5 GB of RAM and the app checks for it?
I know i started the old app on a Nvidia Quadro FX 580 with only 512 MB RAM way back in late 2009.
Anyways i got no spare card that satisfies the RAM-requirements, that is sad.

hi,

is there also a record kept for largest *consecutive* AP? for example3, 7, 11 is a Prime AP but not a consecutive one as there is a prime between 3 and 7. Of course, this would be much harder to look for...

River~~

Regarding extending AP: The search program extends AP found back and forth, so the reported length is the maximal one. For example if the program reports AP25 found, we are sure this is not a part of an AP26.

Search Status:

We are currently up to approximately k=10.6M on both shift=0 and shift=640. Shift=0 encompasses all of the original 6 shifts done 6 years ago. We have now surpassed the original search limits for shift=320 and shift=256, so all tests done at those two shifts (as part of shift=0 tasks) are new work. Double checking on (old-style) shift=0, 64, 128, and 192 continues.

Therefore, 60% of the numbers checked (6 of 10 old-style shifts) in each shift=0 task are new, and of course 100% of the numbers checked in shift=640 tasks are new.

The original search limits can be found in the first post in this thread.
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