# Consecutive Primes in Arithmetic Progression

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 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*+2*b*,*a*+3*b*, ... contains infinitely many primes.

*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:
(10^{10} + 24493 + 30*k*,
*k* = 0, 1, 2, 3, 4). That same year Lander &
Parkin discovered six
(121174811 + 30*k*, *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

rank prime digits who when comment 1 2494779036241 · 2^{49800}+ 1315004 c93 Apr 2022 term 3, difference 6 2 664342014133 · 2^{39840}+ 112005 p408 Apr 2020 term 3, difference 30 3 3428602715439 · 2^{35678}+ 1310753 c93 Apr 2020 term 3, difference 6, ECPP 4 2683143625525 · 2^{35176}+ 1310602 c92 Dec 2019 term 3, difference 6, ECPP 5 1213266377 · 2^{35000}+ 485910546 c4 Mar 2014 ECPP, term 3, difference 2430 6 62399583639 · 9923# - 33994215174285 c98 Nov 2021 term 4, difference 30, ECPP 7 62753735335 · 7919# + 33994216673404 c98 Oct 2021 term 4, difference 30, ECPP 8 (1021328211729 · 2521# · (483 · 2521# + 1) + 2310) · (483 · 2521# - 1)/210 + 193207 c100 Jul 2023 term 4, difference 6, ECPP 9 121152729080 · 7019#/1729 + 193025 c92 Oct 2019 term 4, difference 6, ECPP 10 62037039993 · 7001# + 78115558133021 x38 Oct 2013 term 4, difference 30, ECPP 11 2738129459017 · 4211# + 33994216371805 c98 Jan 2022 term 5, difference 30 12 652229318541 · 3527# + 33994216371504 c98 Oct 2021 term 5, difference 30, ECPP 13 449209457832 · 3307# + 16330504031408 c98 Oct 2021 term 5, difference 30, ECPP 14 2746496109133 · 3001# + 270111290 c97 Oct 2021 term 5, difference 30, ECPP 15 406463527990 · 2801# + 16330504031209 x38 Nov 2013 term 5, difference 30 16 533098369554 · 2357# + 33994216671012 c98 Nov 2021 term 6, difference 30, ECPP

### Related Pages

- The largest known CPAP's of each length by J K Andersen now by Norman Luhn

### 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. NelsonandP. 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. NelsonandP. Zimmermann, "Ten consecutive primes in arithmetic progression,"Math. Comp.,71:239 (2002) 1323--1328 (electronic).MR 1 898 760(Abstract available)- DN97
H. DubnerandH. Nelson, "Seven consecutive primes in arithmetic progression,"Math. Comp.,66(1997) 1743--1749.MR 98a:11122(Abstract available)- 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- 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.]- JLB67
M. F. Jones,M. LalandW. 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. LanderandT. R. Parkin, "Consecutive primes in arithmetic progression,"Math. Comp.,21(1967) 489.- LP67
L. J. LanderandT. R. Parkin, "On first appearance of prime differences,"Math. Comp.,21:99 (1967) 483-488.MR 37:6237

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