Primes in Arithmetic Progression
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Primes in Arithmetic Progression

In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by ${\displaystyle a_{n}=3+4n}$ for ${\displaystyle 0\leq n\leq 2}$.

According to the Green-Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form ${\displaystyle an+b}$, where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites.

For integer k >= 3, an AP-k (also called PAP-k) is any sequence of k primes in arithmetic progression. An AP-k can be written as k primes of the form a·n + b, for fixed integers a (called the common difference) and b, and k consecutive integer values of n. An AP-k is usually expressed with n = 0 to k − 1. This can always be achieved by defining b to be the first prime in the arithmetic progression.

## Properties

Any given arithmetic progression of primes has a finite length. In 2004, Ben J. Green and Terence Tao settled an old conjecture by proving the Green-Tao theorem: The primes contain arbitrarily long arithmetic progressions.[1] It follows immediately that there are infinitely many AP-k for any k.

If an AP-k does not begin with the prime k, then the common difference is a multiple of the primorial k# = 2·3·5·...·j, where j is the largest prime k.

Proof: Let the AP-k be a·n + b for k consecutive values of n. If a prime p does not divide a, then modular arithmetic says that p will divide every p'th term of the arithmetic progression. (From H.J. Weber, Cor.10 in Exceptional Prime Number Twins, Triplets and Multiplets," arXiv:1102.3075[math.NT]. See also Theor.2.3 in Regularities of Twin, Triplet and Multiplet Prime Numbers," arXiv:1103.0447[math.NT], Global J.P.A.Math 8(2012), in press.) If the AP is prime for k consecutive values, then a must therefore be divisible by all primes pk.

This also shows that an AP with common difference a cannot contain more consecutive prime terms than the value of the smallest prime that does not divide a.

If k is prime then an AP-k can begin with k and have a common difference which is only a multiple of (k−1)# instead of k#. (From H. J. Weber, Less Regular Exceptional and Repeating Prime Number Multiplets," arXiv:1105.4092[math.NT], Sect.3.) For example, the AP-3 with primes {3, 5, 7} and common difference 2# = 2, or the AP-5 with primes {5, 11, 17, 23, 29} and common difference 4# = 6. It is conjectured that such examples exist for all primes k. As of 2018, the largest prime for which this is confirmed is k = 19, for this AP-19 found by Wojciech I?ykowski in 2013:

19 + 4244193265542951705·17#·n, for n = 0 to 18.[2]

It follows from widely believed conjectures, such as Dickson's conjecture and some variants of the prime k-tuple conjecture, that if p > 2 is the smallest prime not dividing a, then there are infinitely many AP-(p−1) with common difference a. For example, 5 is the smallest prime not dividing 6, so there is expected to be infinitely many AP-4 with common difference 6, which is called a sexy prime quadruplet. When a = 2, p = 3, it is the twin prime conjecture, with an "AP-2" of 2 primes (b, b + 2).

## Minimal primes in AP

We minimize the last term.[3]

Minimal AP-k
k Primes for n = 0 to k−1
3 3 + 2n
4 5 + 6n
5 5 + 6n
6 7 + 30n
7 7 + 150n
8 199 + 210n
9 199 + 210n
10 199 + 210n
11 110437 + 13860n
12 110437 + 13860n
13 4943 + 60060n
14 31385539 + 420420n
15 115453391 + 4144140n
16 53297929 + 9699690n
17 3430751869 + 87297210n
18 4808316343 + 717777060n
19 8297644387 + 4180566390n
20 214861583621 + 18846497670n
21 5749146449311 + 26004868890n

## Largest known primes in AP

For prime q, q# denotes the primorial 2·3·5·7·...·q.

As of September 2019, the longest known AP-k is an AP-27. Several examples are known for AP-26. The first to be discovered was found on April 12, 2010 by Benoãt Perichon on a PlayStation 3 with software by Jaros?aw Wróblewski and Geoff Reynolds, ported to the PlayStation 3 by Bryan Little, in a distributed PrimeGrid project:[2]

43142746595714191 + 23681770·23#·n, for n = 0 to 25. (23# = 223092870) (sequence in the OEIS)

By the time the first AP-26 was found the search was divided into 131,436,182 segments by PrimeGrid[4] and processed by 32/64bit CPUs, Nvidia CUDA GPUs, and Cell microprocessors around the world.

Before that, the record was an AP-25 found by Raanan Chermoni and Jaros?aw Wróblewski on May 17, 2008:[2]

6171054912832631 + 366384·23#·n, for n = 0 to 24. (23# = 223092870)

The AP-25 search was divided into segments taking about 3 minutes on Athlon 64 and Wróblewski reported "I think Raanan went through less than 10,000,000 such segments"[5] (this would have taken about 57 cpu years on Athlon 64).

The earlier record was an AP-24 found by Jaros?aw Wróblewski alone on January 18, 2007:

468395662504823 + 205619·23#·n, for n = 0 to 23.

For this Wróblewski reported he used a total of 75 computers: 15 64-bit Athlons, 15 dual core 64-bit Pentium D 805, 30 32-bit Athlons 2500, and 15 Durons 900.[6]

The following table shows the largest known AP-k with the year of discovery and the number of decimal digits in the ending prime. Note that the largest known AP-k may be the end of an AP-(k+1). Some record setters choose to first compute a large set of primes of form c·p#+1 with fixed p, and then search for AP's among the values of c that produced a prime. This is reflected in the expression for some records. The expression can easily be rewritten as a·n + b.

Largest known AP-k as of August 2020[2]
k Primes for n = 0 to k−1 Digits Year Discoverer
3 (2723880039837·21290000-1) + (4125·21445205 - 2723880039837·21290000)·n 435054 2016 David Broadhurst, David Abrahmi, David Metcalfe, PrimeGrid
4 (1021747532 + 7399459·n)·60013#+1 25992 2019 Ken Davis
5 (161291608 + 59874860·n)·24001#+1 10378 2018 Ken Davis
6 (1445494494 + 141836149·n)·16301# + 1 7036 2018 Ken Davis
7 (234043271 + 481789017·n)·7001# + 1 3019 2012 Ken Davis
8 (48098104751 + 3026809034·n)·5303# + 1 2271 2019 Norman Luhn, Paul Underwood, Ken Davis
9 (65502205462 + 6317280828·n)·2371# + 1 1014 2012 Ken Davis, Paul Underwood
10 (20794561384 + 1638155407·n)·1050# + 1 450 2019 Norman Luhn
11 (16533786790 + 1114209832·n)·666# + 1 289 2019 Norman Luhn
12 (15079159689 + 502608831·n)·420# + 1 180 2019 Norman Luhn
13 (50448064213 + 4237116495·n)·229# + 1 103 2019 Norman Luhn
14 (55507616633 + 670355577·n)·229# + 1 103 2019 Norman Luhn
15 (14512034548 + 87496195·n)·149# + 1 68 2019 Norman Luhn
16 (9700128038 + 75782144·(n+1))·83# + 1 43 2019 Norman Luhn
17 (9700128038 + 75782144·n)·83# + 1 43 2019 Norman Luhn
18 (33277396902 + 139569962·(n+1))·53# + 1 31 2019 Norman Luhn
19 (33277396902 + 139569962·n)·53# + 1 31 2019 Norman Luhn
20 23 + 134181089232118748020·19#·n 29 2017 Wojciech Izykowski
21 5547796991585989797641 + 29#·n 22 2014 Jaros?aw Wróblewski
22 22231637631603420833 + 8·41#·(n + 1) 20 2014 Jaros?aw Wróblewski
23 22231637631603420833 + 8·41#·n 20 2014 Jaros?aw Wróblewski
24 224584605939537911 + 81292139·23#·(n+3) 18 2019 Rob Gahan, PrimeGrid
25 224584605939537911 + 81292139·23#·(n+2) 18 2019 Rob Gahan, PrimeGrid
26 224584605939537911 + 81292139·23#·(n+1) 18 2019 Rob Gahan, PrimeGrid
27 224584605939537911 + 81292139·23#·n 18 2019 Rob Gahan, PrimeGrid

## Consecutive primes in arithmetic progression

Consecutive primes in arithmetic progression refers to at least three consecutive primes which are consecutive terms in an arithmetic progression. Note that unlike an AP-k, all the other numbers between the terms of the progression must be composite. For example, the AP-3 {3, 7, 11} does not qualify, because 5 is also a prime.

For an integer k >= 3, a CPAP-k is k consecutive primes in arithmetic progression. It is conjectured there are arbitrarily long CPAP's. This would imply infinitely many CPAP-k for all k. The middle prime in a CPAP-3 is called a balanced prime. The largest known as of 2018 has 10546 digits.

The first known CPAP-10 was found in 1998 by Manfred Toplic in the distributed computing project CP10 which was organized by Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann.[7] This CPAP-10 has the smallest possible common difference, 7# = 210. The only other known CPAP-10 as of 2018 was found by the same people in 2008.

If a CPAP-11 exists then it must have a common difference which is a multiple of 11# = 2310. The difference between the first and last of the 11 primes would therefore be a multiple of 23100. The requirement for at least 23090 composite numbers between the 11 primes makes it appear extremely hard to find a CPAP-11. Dubner and Zimmermann estimate it would be at least 1012 times harder than a CPAP-10.[8]

## Minimal consecutive primes in AP

The first occurrence of a CPAP-k is only known for k A006560 in the OEIS).

Minimal CPAP-k[9]
k Primes for n = 0 to k−1
3 3 + 2n
4 251 + 6n
5 9843019 + 30n
6 121174811 + 30n

## Largest known consecutive primes in AP

The table shows the largest known case of k consecutive primes in arithmetic progression, for k = 3 to 10.

Largest known CPAP-k as of January 2020[9]
k Primes for n = 0 to k−1 Digits Year Discoverer
3 2683143625525 · 235176 + 1 + 6n 10602 2019 Gerd Lamprecht, Norman Luhn
4 55072065656 · 7013# + 9843049 + 30n 3024 2018 Gerd Lamprecht
5 2746496109133 · 3001# + 26891 + 30n 1290 2018 Norman Luhn, Gerd Lamprecht
6 386140564676 · 1000# + 26861 + 30n 427 2018 Gerd Lamprecht
7 4785544287883 · 613# + x253 + 210n 266 2007 Jens Kruse Andersen
8 10097274767216 · 250# + x99 + 210n 112 2003 Jens Kruse Andersen
9 73577019188277 · 199#·227·229 + x87 + 210n 101 2005 Hans Rosenthal, Jens Kruse Andersen
10 1180477472752474 · 193# + x77 + 210n 93 2008 Manfred Toplic, CP10 project

xd is a d-digit number used in one of the above records to ensure a small factor in unusually many of the required composites between the primes.
x77 = 54538241683887582 668189703590110659057865934764 604873840781923513421103495579
x87 = 279872509634587186332039135 414046330728180994209092523040 703520843811319320930380677867
x99 = 158794709 618074229409987416174386945728 371523590452459863667791687440 944143462160821328735143564091
x253 = 1617599298905 320471304802538356587398499979 836255156671030473751281181199 911312259550734373874520536148 519300924327947507674746679858 816780182478724431966587843672 408773388445788142740274329621 811879827349575247851843514012 399313201211101277175684636727

## Notes

1. ^ Green, Ben; Tao, Terence (2008), "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics, 167 (2): 481-547, arXiv:math.NT/0404188, doi:10.4007/annals.2008.167.481, MR 2415379
2. ^ a b c d Jens Kruse Andersen, Primes in Arithmetic Progression Records. Retrieved 2020-08-31.
3. ^ OEIS sequence A133277
4. ^ John, AP26 Forum. Retrieved 2013-10-20.
5. ^ Wróblewski, Jaros?aw (2008-05-17). "AP25". primenumbers (Mailing list). Retrieved .
6. ^ Wróblewski, Jaros?aw (2007-01-18). "AP24". primeform (Mailing list). Retrieved .
7. ^ H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, P. Zimmermann, Ten consecutive primes in arithmetic progression, Mathematics of Computation 71 (2002), 1323-1328.
8. ^ Manfred Toplic, The nine and ten primes project. Retrieved on 2007-06-17.
9. ^ a b Jens Kruse Andersen, The Largest Known CPAP's. Retrieved on 2020-01-28.