Cunningham Chains (2nd kind)

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.

Definitions and Notes

A Cunningham chain of length k of the second kind is a sequence of k primes, each which is twice the proceeding one minus one. (For example, {2, 3, 5} and {1531, 3061, 6121, 12241, 24481}.) This means the terms in such a sequence are p, 2p-1, 4p-3, 8p-7, ... so if a prime is labelled "(4p-3)" in the table of primes, it is the third term in a sequence of (at least) three primes.

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.

Record Primes of this Type

rankprime digitswhowhencomment
1213778324725 · 2561418 + 1 169015 p430 Mar 2023 Cunningham chain 2nd kind (2p - 1)
22 · 89396295016384 + 1 146659 p428 Feb 2023 Cunningham chain 2nd kind (2p - 1)
33622179275715 · 2256003 + 1 77078 x47 May 2020 Cunningham chain 2nd kind (2p - 1)
42570606397 · 2252763 + 1 76099 p364 May 2020 Cunningham chain 2nd kind (2p - 1)
52 · 3526667708192 + 1 70021 p409 May 2020 Cunningham chain 2nd kind (2p - 1)
6742478255901 · 240069 + 1 12074 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3)
7996824343 · 240074 + 1 12073 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3)
8198429723072 · 1111005 + 1 11472 L3323 Dec 2016 Cunningham chain 2nd kind (4p - 3)
99649755890145 · 233335 + 1 10048 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3)
1015162914750865 · 233219 + 1 10014 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3)
1149325406476 · 9811# · 8 + 1 4234 p382 Jul 2019 Cunningham chain 2nd kind (8p - 7)
122072453060816 · 7699# + 1 3316 p364 Jun 2019 Cunningham chain 2nd kind (8p - 7)
13138281163736 · 6977# + 1 3006 p395 Jul 2016 Cunningham chain 2nd kind (8p - 7)
14284787490256 · 6701# + 1 2879 p364 Mar 2015 Cunningham chain 2nd kind (8p - 7)
15772463767240 · 5303# + 1 2272 p308 Aug 2019 Cunningham chain 2nd kind (8p - 7)
16102619722624 · 3797# + 1 1631 p395 Sep 2016 Cunningham chain 2nd kind (16p - 15)
17898966996992 · 3001# + 1 1289 p364 Mar 2015 Cunningham chain 2nd kind (16p - 15)
1816 · 2658132486528 · 2969# + 1 1281 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15)
1916 · 1413951139648 · 2969# + 1 1280 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15)
201290733709840 · 2677# + 1 1141 p295 Jan 2011 Cunningham chain 2nd kind (16p - 15)

Weighted Record Primes of this Type

For amusement purposes only we might seek to weight the chains on the list of largest known primes by an estimate of how rare chains of that length are.

We might start with the usual estimate of how hard it is to prove primality of a number the size of n

log(n)2 log log n

and 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.

rankprime digitswhowhencomment
1102619722624 · 3797# + 1 1631 p395 Sep 2016 Cunningham chain 2nd kind (16p - 15)
2898966996992 · 3001# + 1 1289 p364 Mar 2015 Cunningham chain 2nd kind (16p - 15)
316 · 2658132486528 · 2969# + 1 1281 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15)
416 · 1413951139648 · 2969# + 1 1280 p382 Jul 2017 Cunningham chain 2nd kind (16p - 15)
549325406476 · 9811# · 8 + 1 4234 p382 Jul 2019 Cunningham chain 2nd kind (8p - 7)
61290733709840 · 2677# + 1 1141 p295 Jan 2011 Cunningham chain 2nd kind (16p - 15)
72072453060816 · 7699# + 1 3316 p364 Jun 2019 Cunningham chain 2nd kind (8p - 7)
8138281163736 · 6977# + 1 3006 p395 Jul 2016 Cunningham chain 2nd kind (8p - 7)
9284787490256 · 6701# + 1 2879 p364 Mar 2015 Cunningham chain 2nd kind (8p - 7)
10772463767240 · 5303# + 1 2272 p308 Aug 2019 Cunningham chain 2nd kind (8p - 7)
11742478255901 · 240069 + 1 12074 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3)
12996824343 · 240074 + 1 12073 p395 Sep 2016 Cunningham chain 2nd kind (4p - 3)
13198429723072 · 1111005 + 1 11472 L3323 Dec 2016 Cunningham chain 2nd kind (4p - 3)
149649755890145 · 233335 + 1 10048 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3)
1515162914750865 · 233219 + 1 10014 p364 Mar 2015 Cunningham chain 2nd kind (4p - 3)
16213778324725 · 2561418 + 1 169015 p430 Mar 2023 Cunningham chain 2nd kind (2p - 1)
172 · 89396295016384 + 1 146659 p428 Feb 2023 Cunningham chain 2nd kind (2p - 1)
183622179275715 · 2256003 + 1 77078 x47 May 2020 Cunningham chain 2nd kind (2p - 1)
192570606397 · 2252763 + 1 76099 p364 May 2020 Cunningham chain 2nd kind (2p - 1)
202 · 3526667708192 + 1 70021 p409 May 2020 Cunningham chain 2nd kind (2p - 1)

References

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