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 (29571282950 · (2190738 - 1) + 4) · 295369 + 3 86138 p408 Dec 2024 term 2, difference 4 2 (78866031017 · (2166678 - 1) - 4) · 283339 + 1 75274 p408 Nov 2024 term 2, difference 4 3 (16472224158 · (2166678 - 1) - 1) · 283341 + 1 75274 p408 Nov 2024 term 2, difference 6 4 (50573264686 · (2110503 - 1) + 1) · 2110505 + 1 66541 p408 Nov 2024 term 2, difference 6 5 (90704749637 · (2110503 - 1) + 2) · 2110504 + 1 66541 p408 Nov 2024 term 2, difference 4 6 17484430616589 · 254201 + 5 16330 E14 Jun 2024 term 3, difference 6, ECPP 7 2494779036241 · 249800 + 13 15004 c93 Apr 2022 term 3, difference 6 8 664342014133 · 239840 + 1 12005 p408 Apr 2020 term 3, difference 30 9 3428602715439 · 235678 + 13 10753 c93 Apr 2020 term 3, difference 6, ECPP 10 2683143625525 · 235176 + 13 10602 c92 Dec 2019 term 3, difference 6, ECPP 11 35734184537 · 11677#/3 + 9 5002 c98 Jun 2024 term 4, difference 6, ECPP 12 62399583639 · 9923# - 3399421517 4285 c98 Nov 2021 term 4, difference 30, ECPP 13 62753735335 · 7919# + 3399421667 3404 c98 Oct 2021 term 4, difference 30, ECPP 14 (1021328211729 · 2521# · (483 · 2521# + 1) + 2310) · (483 · 2521# - 1)/210 + 19 3207 c100 Jul 2023 term 4, difference 6, ECPP 15 121152729080 · 7019#/1729 + 19 3025 c92 Oct 2019 term 4, difference 6, ECPP 16 2738129459017 · 4211# + 3399421637 1805 c98 Jan 2022 term 5, difference 30 17 652229318541 · 3527# + 3399421637 1504 c98 Oct 2021 term 5, difference 30, ECPP 18 449209457832 · 3307# + 1633050403 1408 c98 Oct 2021 term 5, difference 30, ECPP 19 2746496109133 · 3001# + 27011 1290 c97 Oct 2021 term 5, difference 30, ECPP 20 406463527990 · 2801# + 1633050403 1209 x38 Nov 2013 term 5, difference 30
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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