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Cunningham Chains (2nd kind) |
We have a separate page about Cunningham chains of the first kind. Cunningham chains of both kinds are also called chains of nearly doubled primes.
For any given length k there should be infinitely many chains of length k. In fact the number less than N should be asymptotic to
where![]()
where the sequence Bk begins approximately 1.32032 (k=2), 2.85825, 5.553491, 20.2636, 71.9622, 233.878, 677.356.![]()
rank prime digits who when comment 1 3622179275715 · 2256003 + 1 77078 x47 May 2020 Cunningham chain 2nd kind (2p - 1) 2 2570606397 · 2252763 + 1 76099 p364 May 2020 Cunningham chain 2nd kind (2p - 1) 3 2 · 3526667708192 + 1 70021 p409 May 2020 Cunningham chain 2nd kind (2p - 1) 4 2 · 1031571488192 + 1 65647 p409 May 2020 Cunningham chain 2nd kind (2p - 1) 5 556336461 · 2211356 + 1 63634 L3494 May 2019 Cunningham chain 2nd kind (2p - 1) 6 742478255901 · 240069 + 1 12074 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3) 7 996824343 · 240074 + 1 12073 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3) 8 198429723072 · 1111005 + 1 11472 L3323 Dec 2016 Cunningham chain 2nd kind (4p - 3) 9 9649755890145 · 233335 + 1 10048 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3) 10 15162914750865 · 233219 + 1 10014 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3) 11 49325406476 · 9811# · 8 + 1 4234 p382 Jul 2019 Cunningham chain 2nd kind (8p - 7) 12 2072453060816 · 7699# + 1 3316 p364 Jun 2019 Cunningham chain 2nd kind (8p - 7) 13 138281163736 · 6977# + 1 3006 p395 Jul 2016 Cunningham chain 2nd kind (8p - 7) 14 284787490256 · 6701# + 1 2879 p364 Mar 2015 Cunningham chain 2nd kind (8p - 7) 15 772463767240 · 5303# + 1 2272 p308 Aug 2019 Cunningham chain 2nd kind (8p - 7) 16 102619722624 · 3797# + 1 1631 p395 Sep 2016 Cunningham chain 2nd kind (16p - 15) 17 898966996992 · 3001# + 1 1289 p364 Mar 2015 Cunningham chain 2nd kind (16p - 15) 18 16 · 2658132486528 · 2969# + 1 1281 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15) 19 16 · 1413951139648 · 2969# + 1 1280 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15) 20 1290733709840 · 2677# + 1 1141 p295 Jan 2011 Cunningham chain 2nd kind (16p - 15)
log(n)2 log log nand multiply it by the expected number of potential candidates to test before we find one of length k (by the heuristic estimate above)
log(n)k / Bk.We then take the log one more time to make the numbers nice and small.
rank prime digits who when comment 1 102619722624 · 3797# + 1 1631 p395 Sep 2016 Cunningham chain 2nd kind (16p - 15) 2 898966996992 · 3001# + 1 1289 p364 Mar 2015 Cunningham chain 2nd kind (16p - 15) 3 16 · 2658132486528 · 2969# + 1 1281 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15) 4 16 · 1413951139648 · 2969# + 1 1280 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15) 5 49325406476 · 9811# · 8 + 1 4234 p382 Jul 2019 Cunningham chain 2nd kind (8p - 7) 6 1290733709840 · 2677# + 1 1141 p295 Jan 2011 Cunningham chain 2nd kind (16p - 15) 7 2072453060816 · 7699# + 1 3316 p364 Jun 2019 Cunningham chain 2nd kind (8p - 7) 8 138281163736 · 6977# + 1 3006 p395 Jul 2016 Cunningham chain 2nd kind (8p - 7) 9 284787490256 · 6701# + 1 2879 p364 Mar 2015 Cunningham chain 2nd kind (8p - 7) 10 772463767240 · 5303# + 1 2272 p308 Aug 2019 Cunningham chain 2nd kind (8p - 7) 11 742478255901 · 240069 + 1 12074 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3) 12 996824343 · 240074 + 1 12073 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3) 13 198429723072 · 1111005 + 1 11472 L3323 Dec 2016 Cunningham chain 2nd kind (4p - 3) 14 9649755890145 · 233335 + 1 10048 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3) 15 15162914750865 · 233219 + 1 10014 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3) 16 3622179275715 · 2256003 + 1 77078 x47 May 2020 Cunningham chain 2nd kind (2p - 1) 17 2570606397 · 2252763 + 1 76099 p364 May 2020 Cunningham chain 2nd kind (2p - 1) 18 2 · 3526667708192 + 1 70021 p409 May 2020 Cunningham chain 2nd kind (2p - 1) 19 2 · 1031571488192 + 1 65647 p409 May 2020 Cunningham chain 2nd kind (2p - 1) 20 556336461 · 2211356 + 1 63634 L3494 May 2019 Cunningham chain 2nd kind (2p - 1)
- Cunningham1907
- A. Cunnningham, "On hyper-even numbers and on Fermat's numbers," Proc. Lond. Math. Soc., series 2, 5 (1907) 237--274.
- Guy94 (SectionA7)
- 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.]
- Lehmer1965
- D. H. Lehmer, "On certain chains of primes," Proc. Lond. Math. Soc., series 3, 14a (1965) 183--186. MR 31:2222
- LM1980
- C. Lalout and J. Meeus, "Nearly-doubled primes," J. Recreational Math., 13 (1980-81) 30--35.
- Loh89
- G. Löh, "Long chains of nearly doubled primes," Math. Comp., 53 (1989) 751-759. MR 90e:11015 (Abstract available) [Chains of primes for which each is either twice the proceeding one plus one, or each is either twice the proceeding one minus one. See also [Guy94, section A7].]
- Ribenboim95 (p 333)
- P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
- Yates82
- S. Yates, Repunits and repetends, Star Publishing Co., Inc., Boynton Beach, Florida, 1982. pp. vi+215, MR 83k:10014