Last updated: 23 October 2023
Cunningham Chain records |
Original record list created and maintained by Dirk Augustin.
Overworked by Norman Luhn.
Hosted by Jens Kruse Andersen (since October 2021 by Norman Luhn).
Note:
A large Cunningham Chain (abbreviation: CC) of length n will be also listed in the section(s) for CC of length n-1, n-2, ..., if it is large enough.
A Cunningham Chain is a sequence of nearly doubled primes.
One distinguishes between
Cunningham Chains of length n of the 1st kind: n primes, each which is twice the proceeding one plus one; for example (p, 2p+1, 4p+3, 8p+7) is a CC of length 4 of the 1st kind, if each of the four numbers is prime,
Cunningham Chains of length n of the 2nd kind: n primes, each which is twice the proceeding one minus one; for example (p, 2p-1, 4p-3, 8p-7) is a CC of length 4 of the 2nd kind, if each of the four numbers is prime,
In the following "CC2" stands for "Cunningham Chain of length 2", "CC3" stands for "Cunningham Chain of length 3" and so on.
Only the first member of each CC is shown below!
As far as the programs that were used to find the CC are known to me they are written in brackets!
|
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n | kind | first member of CCn ; ( CC2 1st kind are "Sophie Germain primes" ) | Digits | When? | Discoverer | Record History |
---|---|---|---|---|---|---|
2nd |
2618163402417 ⋅ 21290000 - 1 213778324725 ⋅ 2561417 + 1 |
169015 |
17 Mar 2023 |
Brown, PrimeGrid, TwinGen, LLR Ryan Propper, Serge Batalov |
||
2nd |
1128330746865 ⋅ 266439 - 1 742478255901 ⋅ 240067 + 1 |
12074 |
9 Sep 2016 |
Paridon, NewPGen, OpenPFGW Angel, Augustin, NewPGen, OpenFGW |
||
2nd |
158514759928 ⋅ 8501# - 1 49325406476 ⋅ 9811# + 1 |
4233 |
10 Jul 2019 |
Frank Doornink, mtsieve, OpenPFGW Östlin, NewPGen, OpenPFGW |
||
2nd |
31017701152691334912 ⋅ 4091# - 1 181439827616655015936 ⋅ 4673# + 1 |
2018 |
8 Nov 2016 |
Balyakin Balyakin |
||
2nd |
2799873605326 ⋅ 2371# - 1 52992297065385779421184 ⋅ 1531# + 1 |
668 |
15 May 2015 |
Batalov, NewPGen, OpenPFGW Balyakin |
||
2nd |
82466536397303904 ⋅ 1171# - 1 25802590081726373888 ⋅ 1033# + 1 |
453 |
17 Dec 2015 |
Balyakin Balyakin |
||
2nd |
89628063633698570895360 ⋅ 593# - 1 2373007846680317952 ⋅ 761# + 1 |
337 |
10 Mar 2016 |
Balyakin Balyakin |
||
2nd |
553374939996823808 ⋅ 593# - 1 173129832252242394185728 ⋅ 401# + 1 |
187 |
15 May 2015 |
Balyakin Balyakin |
||
2nd |
3696772637099483023015936 ⋅ 311# - 1 2044300700000658875613184 ⋅ 311# + 1 |
150 |
10 Mar 2016 |
Balyakin Balyakin |
||
2nd |
73853903764168979088206401473739410396455001112581722569026969860983656346568919 ⋅ 151# - 1 341841671431409652891648 ⋅ 311# + 1 |
149 |
3 Nov 2016 |
Primecoin Balyakin |
||
2nd |
288320466650346626888267818984974462085357412586437032687304004479168536445314040 ⋅ 83# - 1 906644189971753846618980352 ⋅ 233# + 1 |
109 |
21 Dec 2013 |
Primecoin Primecoin |
||
2nd |
106680560818292299253267832484567360951928953599522278361651385665522443588804123392 ⋅ 61# - 1 38249410745534076442242419351233801191635692835712219264661912943040353398995076864 ⋅ 47# + 1 |
101 |
12 May 2014 |
Primecoin Primecoin |
||
2nd |
14340319624001770765457042636973902777444526766562062468716553623575545716736 ⋅ 59# - 1 5819411283298069803200936040662511327268486153212216998535044251830806354124236416 ⋅ 47# + 1 |
100 |
16 May 2014 |
Primecoin Primecoin |
||
2nd |
14354792166345299956567113728 ⋅ 43# - 1 67040002730422542592 ⋅ 53# + 1 |
40 |
10 Mar 2016 |
Balyakin Balyakin |
||
2nd |
91304653283578934559359 3081594850735522012277717761 |
28 |
3 Apr 2014 |
Wroblewski Chermoni, Wroblewski |
||
2nd |
2759832934171386593519 1540797425367761006138858881 |
28 |
3 Apr 2014 |
Wroblewski Chermoni, Wroblewski |
||
2nd |
example is unknown 658189097608811942204322721 |
27 |
9 Mar 2014 |
- Chermoni, Wroblewski |
||
2nd |
example is unknown 79910197721667870187016101 |
26 |
23 Mar 2014 |
- Chermoni, Wroblewski |
|
|||||
n | kind | first member of CCn | Digits | When? | Discoverer |
---|---|---|---|---|---|
2nd |
1122659 16651 |
5 |
- |
Lehmer Lehmer |
|
2nd |
19099919 15514861 |
8 |
1980/1981 |
Nelson, Meeus Lalout, Meeus |
|
2nd |
85864769 857095381 |
9 |
1989 |
Loeh Loeh |
|
2nd |
26089808579 205528443121 |
12 |
1989 |
Loeh Loeh |
|
2nd |
665043081119 1389122693971 |
13 |
1989 |
Loeh Loeh |
|
2nd |
554688278429 216857744866621 |
15 |
1989 |
Loeh Loeh |
|
2nd |
4090932431513069 758083947856951 |
15 |
1989 |
Brennen Loeh |
|
2nd |
95405042230542329 107588900851484911 |
18 |
28 Oct 1999 |
Jobling Jobling |
|
2nd |
90616211958465842219 69257563144280941 |
17 |
28 Oct 1999 |
Jobling Jobling |
|
2nd |
810433818265726529159 3203000719597029781 |
19 |
Dec 1997 |
Carmody, Jobling Tony Forbes |
NewPGen:
Sieving program mainly for numbers of the form k ⋅ bn ± 1, k ⋅ p# ± 1, written by Paul Jobling.
click
PFGW:
Prime testing program which allows nearly every form
of numbers, written by Chris Nash and speeded up with
some additional code from G.Woltman and Y.Gallot.
click
PRP:
Probable primality test program for numbers of the form k ⋅ bn ± 1, written by George Woltman.
click
Proth:
Full primality test program for numbers of the form k ⋅ bn ± 1, written by Yves Gallot.
click
GenSv:
A Generic Siever for finding ultra-sparse forms, such as exceptionally long Cunningham Chains.
Used by Phil Carmody and Paul Jobling for their longest records.No URL yet, sorry.
Primo:
Primo is a primality proving program based on the ECPP algorithm:
Elliptic Curve Primality Proving. It will test numbers that are not of any special form.
click
TwinGen:
TwinGen is a program to rapidly presieve a set of candidates of the form k ⋅ 2n ± 1 and
remove those candidates which are composite; written by David Underbakke.
click
LLR:
Takes an input file from Paul Jobling's NewPgen, and proves the
primality of numbers of the form k ⋅ 2n - 1 with k < 2n ;written by Jean Penné.
click
sgsieve:
A specialized sieving program developed by Tom Wu for sieving of Sophie-Germain candidates over a range of k and n values.
It generates NewPGen/LLR-format output.
click
PrimeGrid:
PrimeGrid's primary goal is to bring the excitement of prime finding to the "everyday" computer user.
By simply downloading and installing BOINC and attaching to the PrimeGrid project, participants can choose
from a variety of prime forms to search.
click
Primecoin:
Primecoin, developed by Sunny King, is the first proof-of-work based cryptocurrency that has come up
with any kind of workable solution. The central premise of Primecoin is that, instead of useless SHA256 hashes,
the proof of work protocol would require miners to find long chains of prime numbers (Cunningham chains and BiTwins).
click
Remark:
Sometimes the "old" Primeform was used to test the full primality of the members of a CC,
but I have marked them all with PFGW because PFGW is the successor of Primeform.
Please send any new records, corrections or remarks to dirk.augustin@gmx.de
Please check all primes for full primality before sending them to me.
In addition all submitted primes are always double checked by me with PFGW.
Links:
Links to Chris Caldwell's The Prime Pages:
The Top Twenty: Cunningham Chain (1st kind) (only titanic primes)
The Top Twenty: Cunningham Chain (2nd kind) (only titanic primes)
The Prime Glossary: Cunningham
Chain
Other links about CC's:
Carlos Rivera's The Prime Puzzles & Problem Connection:
Problem 26.- The earliest Cunningham Chains.
Warut Roonguthai's Yves Gallot's Proth.exe and Cunningham Chains.
(archived, last update in 2000)
Eric Weisstein's MathWorld, Cunningham Chain
Wikipedia: Cunningham chain
Pages with similar records:
Tony Forbes'and Norman Luhn Prime k-tuplets click
Henri Lifchitz' BiTwin records click