The Top Twenty--a Prime Page Collection

Arithmetic Progressions of Primes

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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. Comments and suggestions requested.

(up) 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

N_k~D_k*N^2/(2(k-1)(log N)^k)
where
Ugly Hardy-Littlewood type product
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.]

(up) Record Primes of this Type

rankprime digitswhowhencomment
13532235829429 · 21290000 - 1 388342 L3494 Jan 2016 term 3, difference 656129614401 · 21290001
23264529598685 · 21290000 - 1 388342 L3494 Jan 2016 term 3, difference 221561854755 · 21290002
33219238907205 · 21290000 - 1 388342 p199 Sep 2015 term 3, difference 683912203455 · 21290001
42724961865247 · 21290000 - 1 388342 p199 Oct 2015 term 3, difference 222119028825 · 21290002
52339196157659 · 21290000 - 1 388342 p199 Sep 2015 term 3, difference 69718264533 · 21290002
62258157308997 · 21290000 - 1 388342 p199 Oct 2015 term 3, difference 477145708425 · 21290001
71978793358915 · 21290000 - 1 388341 p199 Sep 2015 term 3, difference 10032831585 · 21290001
81829094416835 · 21290000 - 1 388341 L3392 Feb 2015 term 3, difference 160128309135 · 2^1290001 [L3494]
9492590931 · 280000 - 1631979959 · 225001 - 1 24092 p199 Oct 2010 term 4, difference 164196977 · 280000 - 1631979959 · 225000
1016481859 · 35023# - 1 15130 p364 Nov 2015 term 4, difference 4190788 · 35023#
111008075799 · 34687# + 1 15004 p252 Jul 2010 term 4, difference 2571033 · 34687#
12263821581 · 245001 - 487069965 · 225002 - 1 13556 p199 Jun 2010 term 4, difference 87940527 · 245001 - 487069965 · 225001
134103163 · 245007 - 183009063 · 225003 - 1 13556 p199 Jun 2010 term 4, difference 1367721 · 245007 - 183009063 · 225002
141213266377 · 235000 + 4859 10546 c4 Mar 2014 ECPP, consecutive primes term 3, difference 2430
151043085905 · 235000 + 18197 10546 c4 Feb 2014 ECPP, consecutive primes term 3, difference 18198
16109061779 · 235003 + 11855 10545 c4 Feb 2014 ECPP, consecutive primes term 3, difference 5928
17350049825 · 235000 + 7703 10545 c4 Jan 2014 ECPP, consecutive primes term 3, difference 3852
18146462479 · 235001 + 8765 10545 c4 Dec 2013 ECPP, consecutive primes term 3, difference 8766
1916267# · 118917167 - 1 7026 p364 Dec 2015 term 5, difference 18797279 · 16267#
20110650531 · 16267# - 1 7026 p364 Dec 2015 term 5, difference 18149866 · 16267#

(up) 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:

  1. We use the natural log in calculating this weight.
  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
1116040452086 · 2371# + 1 1014 p308 Jan 2012 term 9, difference 6317280828 · 2371#
297336164242 · 2371# + 1 1014 p308 Apr 2013 term 9, difference 6350457699 · 2371#
393537753980 · 2371# + 1 1014 p308 Apr 2013 term 9, difference 3388165411 · 2371#
492836168856 · 2371# + 1 1014 p308 Apr 2013 term 9, difference 127155673 · 2371#
569318339141 · 2371# + 1 1014 p308 Jul 2011 term 9, difference 1298717501 · 2371#
63532235829429 · 21290000 - 1 388342 L3494 Jan 2016 term 3, difference 656129614401 · 21290001
73264529598685 · 21290000 - 1 388342 L3494 Jan 2016 term 3, difference 221561854755 · 21290002
83219238907205 · 21290000 - 1 388342 p199 Sep 2015 term 3, difference 683912203455 · 21290001
92724961865247 · 21290000 - 1 388342 p199 Oct 2015 term 3, difference 222119028825 · 21290002
102339196157659 · 21290000 - 1 388342 p199 Sep 2015 term 3, difference 69718264533 · 21290002
112258157308997 · 21290000 - 1 388342 p199 Oct 2015 term 3, difference 477145708425 · 21290001
121978793358915 · 21290000 - 1 388341 p199 Sep 2015 term 3, difference 10032831585 · 21290001
131829094416835 · 21290000 - 1 388341 L3392 Feb 2015 term 3, difference 160128309135 · 2^1290001 [L3494]
143124777373 · 7001# + 1 3019 p155 Feb 2012 term 7, difference 481789017 · 7001#
152968802755 · 2459# + 1 1057 p155 Apr 2009 term 8, difference 359463429 · 2459#
166179783529 · 2411# + 1 1037 p102 Jun 2003 term 8, difference 176836494 · 2411#
17115248484057 · 2371# + 1 1014 p308 Apr 2013 term 8, difference 7327002535 · 2371#
18113236255068 · 2371# + 1 1014 p308 Apr 2013 term 8, difference 6601354956 · 2371#
19112929231161 · 2371# + 1 1014 p308 Apr 2013 term 8, difference 6982118533 · 2371#
20492590931 · 280000 - 1631979959 · 225001 - 1 24092 p199 Oct 2010 term 4, difference 164196977 · 280000 - 1631979959 · 225000

(up) Related Pages

(up) 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)
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