# Arithmetic Progressions of Primes

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

### Definitions and Notes

Are there infinitely many primes in most arithmetic progressions?  Certainly not if the common difference has a prime factor in common with one of the terms (for example: 6, 9, 12, 15, ...).  In 1837, Dirichlet proved that in all other cases the answer was yes:

Dirichlet's Theorem on Primes in Arithmetic Progressions
If a and b are relatively prime positive integers, then the arithmetic progression a, a+b, a+2b, a+3b, ... contains infinitely many primes.

Recall that the prime number theorem states that for any given n, there are asymptotically n/log n primes less than n.  Similarly it can be proven that the sequence a + k*b (k = 1,2,3,...) contains asymptotically n/(phi(b) log n) primes less than n.  This estimate does not depend on the choice of a!

Dirichlet's theorem does not say that there are arbitrarily many consecutive terms in this sequence which are primes (which is what we'd like).  But Dickson's conjecture does suggests that given any positive integer n, then for each "acceptable" arithmetic progression there are n consecutive terms which are prime.  In 1939, van der Corput showed that there are infinitely many triples of primes in arithmetic progression [Corput1939].  In 2004, Green and Tao [GT2004a] showed that there are indeed arbitrarily long sequences of primes and that a k-term sequence of primes occurs before [GT2004b]:

22222222100k

Obviously this is not optimal!  It is conjectured that it actually occurs before k!+1.

But either way, there is a world of difference between what we know to be true (there are infinitely long arithmetic progressions of primes), and what we have computed: the longest is just over two dozen terms! (See Jens Kruse Andersen's excellent pages linked below.)

It is also possible to put this into a quantitative form and heuristically estimate how many there should be.  For example, Grosswald [GH79] suggested that if Nk is the number of arithmetic progressions of k primes all less than N, then

where
He was able to prove this for the case k=3 [GH79].  Green and Tao have recently proven it for k=4 [GT2006a].

In our heuristics pages we also give asymptotic estimates for the number with fixed length k and fixed difference d.  The first table shows the largest known primes in arithmetic sequence (but just the third term and beyond for each sequence).

[ See all such primes on the list.]

### Record Primes of this Type

rankprime digitswhowhencomment
133 · 22939064 - 5606879602425 · 21290000 - 1 884748 p423 Sep 2021 term 3, difference 33 · 22939063 - 5606879602425 · 21290000
21455 · 22683954 - 6325241166627 · 21290000 - 1 807954 p423 Sep 2021 term 3, difference 1455 · 22683953 - 6325241166627 · 21290000
33555 · 21542813 - 4953427788675 · 21290000 - 1 464437 p363 Apr 2020 term 3, difference 3555 · 21542812 - 4953427788675 · 21290000
44125 · 21445206 - 2723880039837 · 21290000 - 1 435054 p199 Dec 2016 term 3, difference 4125 · 21445205 - 2723880039837 · 21290000
52415 · 21413628 - 1489088842587 · 21290000 - 1 425548 p199 Feb 2017 term 3, difference 2415 · 21413627 - 1489088842587 · 21290000
61524633857 · 299902 - 1 30083 p364 Sep 2022 term 4, difference 928724769 · 299901
722359307 · 60919# + 1 26383 p364 Apr 2022 term 4, difference 5210718 · 60919#
817029817 · 60919# + 1 26383 p364 Apr 2022 term 4, difference 1809778 · 60919#
91043945909 · 60013# + 1 25992 p155 Jul 2019 term 4, difference 7399459 · 60013#
101041073153 · 60013# + 1 25992 p155 May 2019 term 4, difference 10142823 · 60013#
112494779036241 · 249800 + 13 15004 c93 Apr 2022 Consecutive primes term 3, difference 6
12512792361 · 30941# + 1 13338 p364 May 2022 term 5, difference 18195056 · 30941#
13664342014133 · 239840 + 1 12005 p408 Apr 2020 Consecutive primes term 3, difference 30
143428602715439 · 235678 + 13 10753 c93 Apr 2020 Consecutive primes term 3, difference 6, ECPP
152683143625525 · 235176 + 13 10602 c92 Dec 2019 Consecutive primes term 3, difference 6, ECPP
163020616601 · 24499# + 1 10593 p422 Sep 2021 term 6, difference 56497325 · 24499#
172964119276 · 24499# + 1 10593 p422 Sep 2021 term 5, difference 56497325 · 24499#
181213266377 · 235000 + 4859 10546 c4 Mar 2014 ECPP, consecutive primes term 3, difference 2430
19400791048 · 24001# + 1 10378 p155 Nov 2018 term 5, difference 59874860 · 24001#
20393142614 · 24001# + 1 10378 p155 Nov 2018 term 5, difference 54840724 · 24001#

### Weighted Record Primes of this Type

The difficulty of finding such sequences depends on their length.  For example, it will be a long time before an arithmetic sequence of twenty titanic primes is known!  Just for the fun of it, let's attempt to rank these sequences by how long they are.

To rank them, we might take the usual estimate of how hard it is to prove primality of a number the size of n

log(n)2 log log n

and multiply it by the expected number of potential candidates to test before we find one of length k (by the heuristic estimate above):

sqrt(2(k-1)/Dk) (log n)(2+k/2) log log n.

We then take the log one more time just to reduce the size of these numbers.

Notes:

2. The Dk's begin 1.32032363, 2.85824860, 4.15118086, 10.1317949, 17.2986123, and 53.9719483 for k = 3, 4, 5, 6, 7, and 8. They continue 148.551629, 336.034327, 1312.31971, 2364.59896, 7820.60003, 22938.9086, 55651.4626, 91555.1112, 256474.860, 510992.010, 1900972.58, 6423764.31, 18606666.2, 38734732.7, 153217017., 568632503.5, 1941938595 ... [GH79].

rankprime digitswhowhencomment
133 · 22939064 - 5606879602425 · 21290000 - 1 884748 p423 Sep 2021 term 3, difference 33 · 22939063 - 5606879602425 · 21290000
21455 · 22683954 - 6325241166627 · 21290000 - 1 807954 p423 Sep 2021 term 3, difference 1455 · 22683953 - 6325241166627 · 21290000
369285767989 · 5303# + 1 2271 p406 Aug 2019 term 8, difference 3026809034 · 5303#
43020616601 · 24499# + 1 10593 p422 Sep 2021 term 6, difference 56497325 · 24499#
53555 · 21542813 - 4953427788675 · 21290000 - 1 464437 p363 Apr 2020 term 3, difference 3555 · 21542812 - 4953427788675 · 21290000
64125 · 21445206 - 2723880039837 · 21290000 - 1 435054 p199 Dec 2016 term 3, difference 4125 · 21445205 - 2723880039837 · 21290000
72415 · 21413628 - 1489088842587 · 21290000 - 1 425548 p199 Feb 2017 term 3, difference 2415 · 21413627 - 1489088842587 · 21290000
8116040452086 · 2371# + 1 1014 p308 Jan 2012 term 9, difference 6317280828 · 2371#
997336164242 · 2371# + 1 1014 p308 Apr 2013 term 9, difference 6350457699 · 2371#
1093537753980 · 2371# + 1 1014 p308 Apr 2013 term 9, difference 3388165411 · 2371#
1192836168856 · 2371# + 1 1014 p308 Apr 2013 term 9, difference 127155673 · 2371#
1269318339141 · 2371# + 1 1014 p308 Jul 2011 term 9, difference 1298717501 · 2371#
136016459977 · 7927# - 1 3407 p364 Jun 2022 term 7, difference 577051223 · 7927#
142154675239 · 16301# + 1 7036 p155 Apr 2018 term 6, difference 141836149 · 16301#
153124777373 · 7001# + 1 3019 p155 Feb 2012 term 7, difference 481789017 · 7001#
16512792361 · 30941# + 1 13338 p364 May 2022 term 5, difference 18195056 · 30941#
17116814018316 · 5303# + 1 2271 p406 Aug 2019 term 7, difference 10892863626 · 5303#
18116746086504 · 5303# + 1 2271 p406 Aug 2019 term 7, difference 9726011684 · 5303#
19116242725347 · 5303# + 1 2271 p406 Aug 2019 term 7, difference 10388428124 · 5303#
201176100079 · 2591# + 1 1101 p252 Jun 2019 term 8, difference 60355670 · 2591#

### References

BH77
C. Bayes and R. Hudson, "The segmented sieve of Eratosthenes and primes in arithmetic progression," Nordisk Tidskr. Informationsbehandling (BIT), 17:2 (1977) 121--127.  MR 56:5405
Chowla44
S. Chowla, "There exists an infinity of 3--combinations of primes in A. P.," Proc. Lahore Phil. Soc., 6 (1944) 15--16.  MR 7,243l
Corput1939
A. G. van der Corput, "Über Summen von Primzahlen und Primzahlquadraten," Math. Ann., 116 (1939) 1--50.
DN97
H. Dubner and H. Nelson, "Seven consecutive primes in arithmetic progression," Math. Comp., 66 (1997) 1743--1749.  MR 98a:11122 (Abstract available)
GH79
E. Grosswald and P. Hagis, Jr., "Arithmetic progression consisting only of primes," Math. Comp., 33:148 (October 1979) 1343--1352.  MR 80k:10054 (Abstract available)
Grosswald82
E. Grosswald, "Arithmetic progressions that consist only of primes," J. Number Theory, 14 (1982) 9--31.  MR 83k:10081
GT2004a
Green, Ben and Tao, Terence, "The primes contain arbitrarily long arithmetic progressions," Ann. of Math. (2), 167:2 (2008) 481--547.  (http://dx.doi.org/10.4007/annals.2008.167.481) MR 2415379
GT2004b
B. Green and T. Tao, "A bound for progressions of length k in the primes," (2004) Available from http://people.maths.ox.ac.uk/greenbj/papers/back-of-an-envelope.pdf.
GT2006a
Green, Benjamin and Tao, Terence, "Linear equations in primes," Ann. of Math. (2), 171:3 (2010) 1753--1850.  (http://dx.doi.org/10.4007/annals.2010.171.1753) MR 2680398
Guy94 (section A6)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, 1994.  New York, NY, ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
LP67
L. J. Lander and T. R. Parkin, "On first appearance of prime differences," Math. Comp., 21:99 (1967) 483-488.  MR 37:6237
Rose94 (Chpt 13)
H. E. Rose, A course in number theory, second edition, Clarendon Press, 1994.  New York, pp. xvi+398, ISBN 0-19-853479-5; 0-19-852376-9. MR 96g:11001 (Annotation available)