Note: This page was created by Jens Kruse Andersen (2004-2019).
Updated by Norman Luhn (since Oct 2021).
Contact: pzktupel[at]pzktupel[dot].de.

News
2022
May 08: AP25 & AP24 record with 19 digits by cudatail12, PrimeGrid, AP26
April 28. AP4 record with 26383 digits by Serge Batalov
February 8. AP24 record with 19 digits by Kouhki, PrimeGrid, AP26
January 3. AP25 (also AP24) record with 19 digits by Tuna Ertemalp, PrimeGrid, AP26

History Of Additions (2004 - 2021)

Introduction
Submissions

Table for first known AP-k

The largest known AP-k

Smallest AP-k with minimal difference
  Record history for k = 18 to 27

Smallest AP-k with minimal start
  Record history for k = 16 to 27

AP-k with minimal end
  Record history

Credited programs and projects

Links

Introduction
This page shows the largest known case of k primes in AP (arithmetic progression) for each k, with a record history. It also shows the minimal or smallest known difference, start and end in an AP-k, with record histories. Finally it shows all known AP24 until an AP26 was found, and all known AP25. Record histories begin with the record when the page opened in January 2005.
An AP-k is k primes of the form p + d·n for some d (the difference between the primes) and k consecutive values of n. Technically any 2 primes form an AP2 but this page is only about AP3 and longer. Some sources say PAP-k instead of AP-k to signal it is primes in AP.
The Largest Known CPAP's is a different page with records for consecutive primes in AP.

Dirichlet's Theorem on Primes in Arithmetic Progressions says there are always infinitely many primes of the form c + d·n, when c and d are relatively prime - and consecutive n's are not demanded.
Ben Green & Terence Tao presented a proof in 2004 that The primes contain arbitrarily long arithmetic progressions. It shows existence of AP's but does not help in finding them.

k# (called k primorial) is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.
2# = 2, 3# = 6, 5# =30, 7# = 210, 11# = 2310, 13# = 30030, 17# = 510510, 19# = 9699690, 23# = 223092870.
Unless k is prime and starts an AP-k, the difference in an AP-k is always a multiple of k# to avoid factors ≤ k. Expressions with larger primorials are often used in AP searches to avoid more small factors. Most large AP's are found roughly like this:
First compute a large set of primes a·q#+1 for fixed q. Then find AP's among the a's.
This gives an AP formula like (4941928071 + 176836494·n)·2411#+1, for n=0..7
It can be written on the form p + d·n: (4941928071·2411#+1) + (176836494·2411#)·n

Submissions
I would like to hear of all new records. Please mail any you find or know about to jens.k.a@get2net.dk. Say who should be credited, and which program proved the primes if they are above 500 digits.
All primes must be proven, i.e. prp's (probable primes) are not allowed.
The largest prime in an AP determines which AP is judged largest, so an AP3 record could start at 3.
An AP-k is also considered to be an AP-(k-1), so it can hold the record for different k's.

n's not starting at 0 indicates the end of a longer AP. The listed number of digits is for the largest prime in all tables. A year link is to an announcement. The k value links to the record history.

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7, proved for k ≤ 21 as of 2013. The smallest starting prime for the minimal difference is shown below. Finding an AP with a predetermined difference is much harder than an arbitrary difference. Red entries are the smallest known difference when the minimal has not been found.

The minimal starting prime in an AP-k is conjectured to be the smallest prime ≥ k, proved for k ≤ 20 as of 2017. Smaller start primes would lead to a number with a factor ≤ k. The smallest difference for the minimal start is shown below. There may have been earlier discoverers in some cases. Red primes are the smallest known when the minimal is unknown.

Unlike the minimal difference and starting prime, there is probably no system in the minimal ending prime (shown below in parentheses) in an AP-k. Red primes are the smallest known when the minimal is unknown. It appears unlikely there exists a value of k with two AP-k sharing the minimal end, so the table heading does not begin with "Smallest".

Credited programs and projects
Programs:
NewPGen by Paul Jobling.
APSieve by Michael Bell and Jim Fougeron.
APTreeSieve by Jens Kruse Andersen.
srsieve by Geoffrey Reynolds.
TwinGen by David Underbakke.
PSieve by Geoff Reynolds and Ken Brazier.
FermFact by Jim Fougeron.
LLR by Jean Penné.
PrimeForm by the OpenPFGW group with George Woltman.
Proth.exe by Yves Gallot.
PRP by George Woltman.
AP26 by Jaroslaw Wroblewski and Geoff Reynolds.
The primality proving program is only credited above 500 digits.

Projects:
BOINC@Poland (in Polish)
PrimeGrid
Riesel Prime Search
Primeform e-group

Links
From Chris Caldwell's The Prime Pages:
  The Top Twenty: Arithmetic Progressions of Primes (only primes with at least 1000 digits)
  The Prime Glossary: arithmetic sequence
  The Prime Database: Search on Arithmetic
  Dirichlet's Theorem on Primes in Arithmetic Progressions

Records for special AP's:
  Neil Sloane's On-Line Encyclopedia of Integer Sequences: A033189 (start of smallest AP-k with minimal difference), A061558 (smallest difference for AP-k with smallest start), A005115 (smallest end of an AP-k),
  Carlos Rivera's The Prime Puzzles & Problems Connection: Puzzle 34. Prime Triplets in arithmetic progression (largest AP3 starting at 3), and Puzzle 269. 13 primes in A.P. (smallest AP-k starting at k)
  Jens Kruse Andersen: The Largest Known CPAP's (consecutive primes in AP)

Other links:
  Markus Frind: 23 primes in arithmetic progression
  Ben Green & Terence Tao: The primes contain arbitrarily long arithmetic progressions
  Eric Weisstein's MathWorld: Prime Arithmetic Progression
  MathWorld about Green & Tao: Arbitrarily Long Progressions of Primes
  Adrian Chow Ho Yin: Page in chinese I cannot read (may show the first discovered AP12 to AP21)
  Wikipedia: Primes in arithmetic progression
  PrimeGrid: AP26 Search (a distributed project to find the longest AP)

This page is at http://primerecords.dk/aprecords.htm and licensed under the GFDL.
Created and maintained by Jens Kruse Andersen, jens.k.a@get2net.dk   home
Some of the data on minimal difference/start/end AP-k supplied by Gennady Gusev.