# Consecutive Primes in Arithmetic Progression

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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.

### Definitions and Notes

Are there primes in every arithmetic progression? If so, how many?  Dirichlet's theorem tells that the answers are usually 'yes,' and 'there are infinitely many primes.'
Dirichlet's Theorem on Primes in Arithmetic Progressions (1837)
If a and b are relatively prime positive integers, then the arithmetic progression a, a+b, a+2b, a+3b, ... contains infinitely many primes.
This theorem does not say that there are infinitely may consecutive terms in this sequence which are primes. First van der Corput (in 1939) and then Chowla (in 1944) proved this for the case of three consecutive terms.  Finally, in 2004, Ben Green and Terence Tao proved that there were arbitrarily long arithmetic progressions of primes. Here though we have an even more stringent condition. We are looking for n consecutive primes in arithmetic progressions.  It is conjectured that there are such primes, but this has not even been shown in for the case of n=3 primes.

In 1967, Jones, Lal & Blundon found five consecutive primes in arithmetic progression: (1010 + 24493 + 30k, k = 0, 1, 2, 3, 4).  That same year Lander & Parkin discovered six (121174811 + 30k, k = 0, 1, ..., 5).  After a gap of twenty years the number was increased from six to seven by Dubner & Nelson; then in quick succession, eight, nine and finally ten by Dubner, Forbes, Lygeros, Mizony, Nelson & Zimmermann.  They wrote:

In the search for nine and ten primes, we obtained help from the Internet community and by an incredible coincidence the actual discoverer was the same person in both instances - Manfred Toplic.
Those holding the current record of ten expect that the ten-primes record will stand for a long time.  Eleven consecutive primes in arithmetic progression require a common difference of at least 2310 and they project that a search is not feasible without a new idea or a trillion-fold improvement in computer speeds.

### Record Primes of this Type

rankprime digitswhowhencomment
11213266377 · 235000 + 4859 10546 c4 Mar 2014 ECPP, term 3, difference 2430
21043085905 · 235000 + 18197 10546 c4 Feb 2014 ECPP, term 3, difference 18198
3109061779 · 235003 + 11855 10545 c4 Feb 2014 ECPP, term 3, difference 5928
4350049825 · 235000 + 7703 10545 c4 Jan 2014 ECPP, term 3, difference 3852
5146462479 · 235001 + 8765 10545 c4 Dec 2013 ECPP, term 3, difference 8766
662037039993 · 7001# + 7811555813 3021 x38 Oct 2013 term 4, difference 30, ECPP
750946848056 · 7001# + 7811555813 3021 x38 Oct 2013 term 4, difference 30, ECPP
826997933312 · 7001# + 7811555753 3020 x38 Oct 2013 term 4, difference 30, ECPP
925506692100 · 7001# + 7811555783 3020 x38 Oct 2013 term 4, difference 30, ECPP
10198267970563 · 6007# + 7811555753 2575 x38 Oct 2013 term 4, difference 30, ECPP
11406463527990 · 2801# + 1633050403 1209 x38 Nov 2013 term 5, difference 30
12993530619517 · 2503# + 1633050373 1073 x38 Dec 2013 term 5, difference 30
13495690450643 · 2503# + 1633050403 1072 x38 Nov 2013 term 5, difference 30
14150822742857 · 2503# + 1633050373 1072 x38 Nov 2013 term 5, difference 30
1594807777362 · 2503# + 1633050373 1072 x38 Nov 2013 term 5, difference 30

### References

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.
DFLMNZ1998
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson and P. Zimmermann, "Ten consecutive primes in arithmetic progression," Math. Comp., 71:239 (2002) 1323--1328 (electronic).  MR 1 898 760 (Abstract available)
DFLMNZ1998
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson and P. Zimmermann, "Ten consecutive primes in arithmetic progression," Math. Comp., 71:239 (2002) 1323--1328 (electronic).  MR 1 898 760 (Abstract available)
DN97
H. Dubner and H. Nelson, "Seven consecutive primes in arithmetic progression," Math. Comp., 66 (1997) 1743--1749.  MR 98a:11122 (Abstract available)
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
Guy94 (section A6)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, NY, 1994.  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.]
JLB67
M. F. Jones, M. Lal and W. J. Blundon, "Statistics on certain large primes," Math. Comp., 21:97 (1967) 103--107.  MR 36:3707
Kra2005
B. Kra, "The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view," Bull. Amer. Math. Soc., 43:1 (2006) 3--23 (electronic).  (http://dx.doi.org/10.1090/S0273-0979-05-01086-4) MR 2188173 (Abstract available)
LP1967a
L. J. Lander and T. R. Parkin, "Consecutive primes in arithmetic progression," Math. Comp., 21 (1967) 489.
LP67
L. J. Lander and T. R. Parkin, "On first appearance of prime differences," Math. Comp., 21:99 (1967) 483-488.  MR 37:6237