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+2b, a+3b, ... contains infinitely many 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 (1021328211729 · 2521# · (483 · 2521# + 1) + 2310) · (483 · 2521# - 1)/210 + 19 3207 c100 Jul 2023 term 4, difference 6, ECPP 9 121152729080 · 7019#/1729 + 19 3025 c92 Oct 2019 term 4, difference 6, ECPP 10 62037039993 · 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
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. 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, 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. 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
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