A Cunningham chain of the first kind of length n is a sequence of prime numbers (p1, ..., pn) such that for all 1 ≤ i < n, pi+1 = 2pi + 1. (Hence each term of such a chain except the last one is a Sophie Germain prime, and each term except the first is a safe prime).
It follows that
Or, by setting (the number is not part of the sequence and need not be a prime number), we have
Similarly, a Cunningham chain of the second kind of length n is a sequence of prime numbers (p1,...,pn) such that for all 1 ≤ i < n, pi+1 = 2pi − 1.
It follows that the general term is
Now, by setting , we have .
Cunningham chains are also sometimes generalized to sequences of prime numbers (p1, ..., pn) such that for all 1 ≤ i ≤ n, pi+1 = api + b for fixed coprime integers a, b; the resulting chains are called generalized Cunningham chains.
A Cunningham chain is called complete if it cannot be further extended, i.e., if the previous or next term in the chain would not be a prime number anymore.
Examples of complete Cunningham chains of the first kind include these:
- 2, 5, 11, 23, 47 (The next number would be 95, but that is not prime.)
- 3, 7 (The next number would be 15, but that is not prime.)
- 29, 59 (The next number would be 119, but that is not prime.)
- 41, 83, 167 (The next number would be 335, but that is not prime.)
Examples of complete Cunningham chains of the second kind include these:
- 2, 3, 5 (The next number would be 9, but that is not prime.)
- 7, 13 (The next number would be 25, but that is not prime.)
- 19, 37, 73 (The next number would be 145, but that is not prime.)
- 31, 61 (The next number would be 121, but that is not prime.)
Cunningham chains are now considered useful in cryptographic systems since "they provide two concurrent suitable settings for the ElGamal cryptosystem ... [which] can be implemented in any field where the discrete logarithm problem is difficult."
Largest known Cunningham chains
It follows from Dickson's conjecture and the broader Schinzel's hypothesis H, both widely believed to be true, that for every k there are infinitely many Cunningham chains of length k. There are, however, no known direct methods of generating such chains.
|k||Kind||p1 (starting prime)||Digits||Year||Discoverer|
|1||257885161 − 1||17425170||2013||Curtis Cooper, GIMPS|
|2||1st||18543637900515×2666667 − 1||200701||2012||Philipp Bliedung, PrimeGrid|
|2nd||648309×2148310 + 1||44652||2010||Tom Wu|
|3||1st||914546877×234772 − 1||10477||2010||Tom Wu|
|2nd||82659189×226997 + 1||8135||2010||Tom Wu|
|4||1st||1249097877×6599# − 1||2835||2011||Michael Angel|
|2nd||630698711×4933# + 1||2105||2010||Michael Angel|
|5||1st||4250172704×2749# − 1||1183||2012||Dirk Augustin|
|2nd||80670856865×2677# + 1||1140||2011||Michael Angel|
|6||1st||37488065464×1483# − 1||633||2010||Dirk Augustin|
|2nd||37783362904×1097# + 1||475||2006||Dirk Augustin|
|7||1st||162597166369×827# − 1||356||2010||Dirk Augustin|
|2nd||668302064×593# + 786153598231||251||2008||Thomas Wolter & Jens Kruse Andersen|
|8||1st||2×65728407627×431# − 1||186||2005||Dirk Augustin|
|2nd||1148424905221×509# + 1||224||2010||Dirk Augustin|
|9||1st||65728407627×431# − 1||185||2005||Dirk Augustin|
|2nd||182887101390961871050645934589918687746535370612015546956692154622371784133412186×223# + 1||167||2013||Primecoin (block 79349)|
|10||1st||44598464649019035883154084128331646888059795218766083584048621139159337786287845212160000×149# − 1||146||2013||Primecoin (block 182690)|
|2nd||2361221366027763072635481564211745987513418780208430432809344944242307595601310582×139# + 1||137||2013||Primecoin (block 125775)|
|11||1st||73853903764168979088206401473739410396455001112581722569026969860983656346568919×151# − 1||140||2013||Primecoin (block 95569)|
|2nd||8026337833619599372491948674562462668692014872229571339857384053514279156849912832×109# + 1||127||2014||Primecoin (block 365304)|
|12||1st||61592551716229060392971860549140211602858978086524024531871935735163762961673908480×71# − 1||110||2013||Primecoin (block 239833)|
|2nd||160433998429454286861864982184342218645773889300991352796925862298096263175269000×73# + 1||109||2013||Primecoin (block 323183)|
|13||1st||106680560818292299253267832484567360951928953599522278361651385665522443588804123392×61# − 1||107||2014||Primecoin (block 368051)|
|2nd||457905006824220038355933583213167637693837605663532362031057034656375497233892960×37# + 1||94||2014||Primecoin (block 375981)|
|14||1st||9510321949318457733566099||25||2004||Jens Kruse Andersen|
|2nd||335898524600734221050749906451371||33||2008||Jens Kruse Andersen|
q# denotes the primorial 2×3×5×7×...×q.
As of 2013[update], the longest known Cunningham chain of either kind is of length 17. The first known was of the 1st kind starting at 2759832934171386593519, discovered by Jaroslaw Wroblewski in 2008 where he also found some of the 2nd kind.
Congruences of Cunningham chains
Let the odd prime be the first prime of a Cunningham chain of the first kind. The first prime is odd, thus . Since each successive prime in the chain is it follows that . Thus, , , and so forth.
The above property can be informally observed by considering the primes of a chain in base 2. (Note that, as with all bases, multiplying by the number of the base "shifts" the digits to the left.) When we consider in base 2, we see that, by multiplying by 2, the least significant digit of becomes the secondmost least significant digit of . Because is odd—that is, the least significant digit is 1 in base 2--we know that the secondmost least significant digit of is also 1. And, finally, we can see that will be odd due to the addition of 1 to . In this way, successive primes in a Cunningham chain are essentially shifted left in binary with ones filling in the least significant digits. For example, here is a complete length 6 chain which starts at 141361469:
A similar result holds for Cunningham chains of the second kind. From the observation that and the relation it follows that . In binary notation, the primes in a Cunningham chain of the second kind end with a pattern "0...01", where, for each , the number of zeros in the pattern for is one more than the number of zeros for . As with Cunningham chains of the first kind, the bits left of the pattern shift left by one position with each successive prime.
- Joe Buhler, Algorithmic Number Theory: Third International Symposium, ANTS-III. New York: Springer (1998): 290
- Dirk Augustin, Cunningham Chain records. Retrieved on 2013-12-14.
- Löh, Günter (October 1989). "Long chains of nearly doubled primes". Mathematics of Computation 53 (188): 751–759. doi:10.1090/S0025-5718-1989-0979939-8.
- Primecoin, which uses Cunningham chains as a proof-of-work system
- The Prime Glossary: Cunningham chain
- PrimeLinks++: Cunningham chain
- Sequence A005602 in OEIS: the first term of the lowest complete Cunningham chains of the first kind of length n, for 1 ≤ n ≤ 14
- Sequence A005603 in OEIS: the first term of the lowest complete Cunningham chains of the second kind with length n, for 1 ≤ n ≤ 15