# 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.

This page is about one of those forms.

### 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
aandbare relatively prime positive integers, then the arithmetic progressiona,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]:

2^{2222222100k}

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 *N _{k}* 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

rank prime digits who when comment 1 1103 · 2^{3558177}- 503 · 2^{1092022}- 11071122 p423 Dec 2022 term 3, difference 1103 · 2 ^{3558176}- 503 · 2^{1092022}2 2109 · 2^{3423798}- 3027 · 2^{988658}+ 11030670 CH13 Jan 2023 term 3, difference 2109 · 2 ^{3423797}- 3027 · 2^{988658}3 2895 · 2^{3422031}- 143157 · 2^{2144728}+ 11030138 p423 Jan 2023 term 3, difference 2895 · 2 ^{3422030}- 143157 · 2^{2144728}4 33 · 2^{2939064}- 5606879602425 · 2^{1290000}- 1884748 p423 Sep 2021 term 3, difference 33 · 2 ^{2939063}- 5606879602425 · 2^{1290000}5 1455 · 2^{2683954}- 6325241166627 · 2^{1290000}- 1807954 p423 Sep 2021 term 3, difference 1455 · 2 ^{2683953}- 6325241166627 · 2^{1290000}6 1524633857 · 2^{99902}- 130083 p364 Sep 2022 term 4, difference 928724769 · 2 ^{99901}7 22359307 · 60919# + 126383 p364 Apr 2022 term 4, difference 5210718 · 60919# 8 17029817 · 60919# + 126383 p364 Apr 2022 term 4, difference 1809778 · 60919# 9 1043945909 · 60013# + 125992 p155 Jul 2019 term 4, difference 7399459 · 60013# 10 1041073153 · 60013# + 125992 p155 May 2019 term 4, difference 10142823 · 60013# 11 2494779036241 · 2^{49800}+ 1315004 c93 Apr 2022 Consecutive primes term 3, difference 6 12 512792361 · 30941# + 113338 p364 May 2022 term 5, difference 18195056 · 30941# 13 664342014133 · 2^{39840}+ 112005 p408 Apr 2020 Consecutive primes term 3, difference 30 14 3428602715439 · 2^{35678}+ 1310753 c93 Apr 2020 Consecutive primes term 3, difference 6, ECPP 15 2683143625525 · 2^{35176}+ 1310602 c92 Dec 2019 Consecutive primes term 3, difference 6, ECPP 16 3020616601 · 24499# + 110593 p422 Sep 2021 term 6, difference 56497325 · 24499# 17 2964119276 · 24499# + 110593 p422 Sep 2021 term 5, difference 56497325 · 24499# 18 1213266377 · 2^{35000}+ 485910546 c4 Mar 2014 ECPP, consecutive primes term 3, difference 2430 19 400791048 · 24001# + 110378 p155 Nov 2018 term 5, difference 59874860 · 24001# 20 393142614 · 24001# + 110378 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 logn

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)/D) (log_{k}n)^{(2+k/2)}log logn.

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

Notes:

- We use the natural log in calculating this weight.
- The
*D*'s begin 1.32032363, 2.85824860, 4.15118086, 10.1317949, 17.2986123, and 53.9719483 for_{k}*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].

rank prime digits who when comment 1 1103 · 2^{3558177}- 503 · 2^{1092022}- 11071122 p423 Dec 2022 term 3, difference 1103 · 2 ^{3558176}- 503 · 2^{1092022}2 2109 · 2^{3423798}- 3027 · 2^{988658}+ 11030670 CH13 Jan 2023 term 3, difference 2109 · 2 ^{3423797}- 3027 · 2^{988658}3 2895 · 2^{3422031}- 143157 · 2^{2144728}+ 11030138 p423 Jan 2023 term 3, difference 2895 · 2 ^{3422030}- 143157 · 2^{2144728}4 33 · 2^{2939064}- 5606879602425 · 2^{1290000}- 1884748 p423 Sep 2021 term 3, difference 33 · 2 ^{2939063}- 5606879602425 · 2^{1290000}5 1455 · 2^{2683954}- 6325241166627 · 2^{1290000}- 1807954 p423 Sep 2021 term 3, difference 1455 · 2 ^{2683953}- 6325241166627 · 2^{1290000}6 69285767989 · 5303# + 12271 p406 Aug 2019 term 8, difference 3026809034 · 5303# 7 3020616601 · 24499# + 110593 p422 Sep 2021 term 6, difference 56497325 · 24499# 8 116040452086 · 2371# + 11014 p308 Jan 2012 term 9, difference 6317280828 · 2371# 9 97336164242 · 2371# + 11014 p308 Apr 2013 term 9, difference 6350457699 · 2371# 10 93537753980 · 2371# + 11014 p308 Apr 2013 term 9, difference 3388165411 · 2371# 11 92836168856 · 2371# + 11014 p308 Apr 2013 term 9, difference 127155673 · 2371# 12 69318339141 · 2371# + 11014 p308 Aug 2011 term 9, difference 1298717501 · 2371# 13 6016459977 · 7927# - 13407 p364 Jun 2022 term 7, difference 577051223 · 7927# 14 2154675239 · 16301# + 17036 p155 Apr 2018 term 6, difference 141836149 · 16301# 15 3124777373 · 7001# + 13019 p155 Feb 2012 term 7, difference 481789017 · 7001# 16 512792361 · 30941# + 113338 p364 May 2022 term 5, difference 18195056 · 30941# 17 116814018316 · 5303# + 12271 p406 Aug 2019 term 7, difference 10892863626 · 5303# 18 116746086504 · 5303# + 12271 p406 Aug 2019 term 7, difference 9726011684 · 5303# 19 116242725347 · 5303# + 12271 p406 Aug 2019 term 7, difference 10388428124 · 5303# 20 1176100079 · 2591# + 11101 p252 Jun 2019 term 8, difference 60355670 · 2591#

### Related Pages

- Jens Kruse Andersen's excellent Primes in Arithmetic Progression Records

### References

- BH77
C. BayesandR. 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. DubnerandH. Nelson, "Seven consecutive primes in arithmetic progression,"Math. Comp.,66(1997) 1743--1749.MR 98a:11122(Abstract available)- GH79
E. GrosswaldandP. 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, BenandTao, 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. GreenandT. Tao, "A bound for progressions of lengthkin the primes," (2004) Available from http://people.maths.ox.ac.uk/greenbj/papers/back-of-an-envelope.pdf.- GT2006a
Green, BenjaminandTao, 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. LanderandT. 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, New York, 1994. pp. xvi+398, ISBN 0-19-853479-5; 0-19-852376-9.MR 96g:11001(Annotation available)