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Consecutive Primes in Arithmetic Progression |
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.
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
- 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