# Cunningham Chains (1st 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.

This page is about one of those forms.

### Definitions and Notes

Recall that a Sophie Germain prime is a prime*p*such that

*q*=2

*p*+1 is also prime. Why not also ask that

*r*=2

*q*+1 is prime, and 2

*r*+1 is prime, and...? A

**Cunningham chain of length**is a sequence of

*k*of the first kind*k*primes, each which is twice the proceeding one plus one. For example, {2, 5, 11, 23, 47} and {89, 179, 359, 719, 1439, 2879}. This means the terms in such a sequence are

*p*, 2

*p*+1, 4

*p*+3, 8

*p*+7, ... so if a prime is labelled "(4

*p*+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 the Cunninham chains of the second kind and the Sophie Germain primes. 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

*B*begins approximately 1.32032 (

_{k}*k*=2), 2.85825, 5.553491, 20.2636, 71.9622, 233.878, 677.356.

Because the Sophie Germain primes have their own Top Twenty page, they are not included in the list of records below. Forthe same reason 'Cunningham Chain (p)'s and 'Cunningham Chain (2p+1)'s (which are Sophie Germain primes) are omitted.

### Record Primes of this Type

rank prime digits who when comment 1 1128330746865 · 2^{66441}- 120013 p158 Feb 2020 Cunningham chain (4p + 3) 2 1815615642825 · 2^{44046}- 113272 p395 Nov 2016 Cunningham chain (4p + 3) 3 778965587811 · 2^{36627}- 111038 p395 Sep 2016 Cunningham chain (4p + 3) 4 272879344275 · 2^{36622}- 111036 p395 Aug 2016 Cunningham chain (4p + 3) 5 333645655005 · 2^{35549}- 110713 p364 Mar 2015 Cunningham chain (4p + 3) 6 1268118079424 · 8501# - 13640 p434 Oct 2023 Cunningham chain (8p + 7) 7 109766820328 · 7877# - 13385 p395 Nov 2016 Cunningham chain (8p + 7) 8 231692481512 · 7517# - 13218 p395 Sep 2016 Cunningham chain (8p + 7) 9 375967981369 · 6907# · 8 - 12972 p382 May 2017 Cunningham chain (8p + 7) 10 354362289656 · 6907# · 8 - 12972 p382 May 2017 Cunningham chain (8p + 7) 11 7610828704751636272 · 4679# - 12020 p151 Feb 2024 Cunningham chain (16p + 15) 12 54201838768 · 3917# - 11681 p395 Oct 2016 Cunningham chain (16p + 15) 13 3199190962192 · 3499# - 11494 p382 Mar 2016 Cunningham chain (16p + 15) 14 5780736564512 · 3023# - 11301 p364 Mar 2015 Cunningham chain (16p + 15) 15 1542946580224 · 2851# - 11231 p364 Aug 2023 Cunningham chain (16p + 15) 16 89595955370432 · 2371# - 11017 p364 Mar 2015 Cunningham chain (32p + 31)

### 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 also include the Sophie Germain primes because they are chains of length two.To form a weight we start with the
usual estimate of how hard it is to prove primality of a
number the size of *n*

log(and multiply it by the expected number of potential candidates to test before we find one of lengthn)^{2}log logn

*k*(by the heuristic estimate above)

log(We then take the log one more time to make the numbers nice and small.n)^{k}/B._{k}

(Because the Sophie germain primes have their own Top Twenty page, they are not included in the list of records below.)

rank prime digits who when comment 1 89595955370432 · 2371# - 11017 p364 Mar 2015 Cunningham chain (32p + 31) 2 7610828704751636272 · 4679# - 12020 p151 Feb 2024 Cunningham chain (16p + 15) 3 54201838768 · 3917# - 11681 p395 Oct 2016 Cunningham chain (16p + 15) 4 3199190962192 · 3499# - 11494 p382 Mar 2016 Cunningham chain (16p + 15) 5 5780736564512 · 3023# - 11301 p364 Mar 2015 Cunningham chain (16p + 15) 6 1542946580224 · 2851# - 11231 p364 Aug 2023 Cunningham chain (16p + 15) 7 1268118079424 · 8501# - 13640 p434 Oct 2023 Cunningham chain (8p + 7) 8 109766820328 · 7877# - 13385 p395 Nov 2016 Cunningham chain (8p + 7) 9 231692481512 · 7517# - 13218 p395 Sep 2016 Cunningham chain (8p + 7) 10 375967981369 · 6907# · 8 - 12972 p382 May 2017 Cunningham chain (8p + 7) 11 354362289656 · 6907# · 8 - 12972 p382 May 2017 Cunningham chain (8p + 7) 12 1128330746865 · 2^{66441}- 120013 p158 Feb 2020 Cunningham chain (4p + 3) 13 1815615642825 · 2^{44046}- 113272 p395 Nov 2016 Cunningham chain (4p + 3) 14 778965587811 · 2^{36627}- 111038 p395 Sep 2016 Cunningham chain (4p + 3) 15 272879344275 · 2^{36622}- 111036 p395 Aug 2016 Cunningham chain (4p + 3) 16 333645655005 · 2^{35549}- 110713 p364 Mar 2015 Cunningham chain (4p + 3)

### Related Pages

- Dirk Augustin's Cunningham Chain Records
- The Top 20 Cunningham Chains of the Second Kind
- The Cunningham Chain's entry from the Prime Glossary

### 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, 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.]- Lehmer1965
D. H. Lehmer, "On certain chains of primes,"Proc. Lond. Math. Soc., series 3,14a(1965) 183--186.MR 31:2222- LM1980
C. LaloutandJ. 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