The Top Twenty--a Prime Page Collection

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.

(up) 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.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1197418203 · 225000 + 6089 7535 FE4 Feb 2005 ECPP, term 3, difference 6090
287 · 224582 + 2579 7402 c31 Nov 2004 ECPP, term 3, difference 1290
34811 · 220219 + 1 6091 DM Oct 1996 term 3, difference 3738 [c36]
4(84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1)/35 + 13 5132 p179 Apr 2006 term 3, difference 6
5(61310346529 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1)/35 + 13 5132 p179 Oct 2005 term 3, difference 6
625885133741 · 5003# + 3399421607 2148 c14 Dec 2012 term 4, difference 30
725900 + 469721940591 1777 c45 Nov 2007 term 4, difference 2880, ECPP
818672891658 · 4099# + 1591789579 1763 c14 Oct 2003 ECPP, term 4, difference 210
923963 + 1031392866 1312 c32 Oct 2005 term 4, difference 1500
104919761805 · 2999# + 6763 1284 c23 Sep 2003 term 4, difference 30
11142661157626 · 2411# + 71427877 1038 c14 May 2002 term 5, difference 30

(up) Related Pages

(up) 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
B. Green and T. Tao, "The primes contain arbitrarily long arithmetic progressions," Ann. Math., preprint. Available from http://arxiv.org/abs/math/0404188v6.
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
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