Consecutive Primes in Arithmetic Progression

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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

rank prime digits who when comment

1 2494779036241 · 2⁴⁹⁸⁰⁰ + 13 15004 c93 Apr 2022 term 3, difference 6
2 664342014133 · 2³⁹⁸⁴⁰ + 1 12005 p408 Apr 2020 term 3, difference 30
3 3428602715439 · 2³⁵⁶⁷⁸ + 13 10753 c93 Apr 2020 term 3, difference 6, ECPP
4 2683143625525 · 2³⁵¹⁷⁶ + 13 10602 c92 Dec 2019 term 3, difference 6, ECPP
5 1213266377 · 2³⁵⁰⁰⁰ + 4859 10546 c4 Mar 2014 ECPP, term 3, difference 2430
6 62399583639 · 9923# - 3399421517 4285 c98 Nov 2021 term 4, difference 30, ECPP
7 62753735335 · 7919# + 3399421667 3404 c98 Oct 2021 term 4, difference 30, ECPP
8 121152729080 · 7019#/1729 + 19 3025 c92 Oct 2019 term 4, difference 6, ECPP
9 62037039993 · 7001# + 7811555813 3021 x38 Oct 2013 term 4, difference 30, ECPP
10 50946848056 · 7001# + 7811555813 3021 x38 Oct 2013 term 4, difference 30, ECPP
11 2738129459017 · 4211# + 3399421637 1805 c98 Jan 2022 term 5, difference 30
12 652229318541 · 3527# + 3399421637 1504 c98 Oct 2021 term 5, difference 30, ECPP
13 449209457832 · 3307# + 1633050403 1408 c98 Oct 2021 term 5, difference 30, ECPP
14 2746496109133 · 3001# + 27011 1290 c97 Oct 2021 term 5, difference 30, ECPP
15 406463527990 · 2801# + 1633050403 1209 x38 Nov 2013 term 5, difference 30
16 533098369554 · 2357# + 3399421667 1012 c98 Nov 2021 term 6, difference 30, ECPP

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

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