## Consecutive Primes in Arithmetic Progression |

**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.

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.

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 121152729080 · 7019#/1729 + 193025 c92 Oct 2019 term 4, difference 6, ECPP 9 62037039993 · 7001# + 78115558133021 x38 Oct 2013 term 4, difference 30, ECPP 10 50946848056 · 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

- The largest known CPAP's of each length by Jens Kruse Andersen

- 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, 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. 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

Chris K. Caldwell
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