## Consecutive Primes in Arithmetic Progression |

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.'
*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.

**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 664342014133 · 2^{39840}+ 112005 p408 Apr 2020 term 3, difference 30 2 3428602715439 · 2^{35678}+ 1310753 c93 Apr 2020 term 3, difference 6, ECPP 3 2683143625525 · 2^{35176}+ 1310602 c92 Dec 2019 term 3, difference 6, ECPP 4 1213266377 · 2^{35000}+ 485910546 c4 Mar 2014 ECPP, term 3, difference 2430 5 1043085905 · 2^{35000}+ 1819710546 c4 Feb 2014 ECPP, term 3, difference 18198 6 121152729080 · 7019#/1729 + 193025 c92 Oct 2019 term 4, difference 6, ECPP 7 62037039993 · 7001# + 78115558133021 x38 Oct 2013 term 4, difference 30, ECPP 8 50946848056 · 7001# + 78115558133021 x38 Oct 2013 term 4, difference 30, ECPP 9 26997933312 · 7001# + 78115557533020 x38 Oct 2013 term 4, difference 30, ECPP 10 25506692100 · 7001# + 78115557833020 x38 Oct 2013 term 4, difference 30, ECPP 11 406463527990 · 2801# + 16330504031209 x38 Nov 2013 term 5, difference 30 12 993530619517 · 2503# + 16330503731073 x38 Dec 2013 term 5, difference 30 13 495690450643 · 2503# + 16330504031072 x38 Nov 2013 term 5, difference 30 14 150822742857 · 2503# + 16330503731072 x38 Nov 2013 term 5, difference 30 15 94807777362 · 2503# + 16330503731072 x38 Nov 2013 term 5, difference 30

- 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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