A Cunningham chain of the first kind of length n is a sequence of prime numbers (p1, ..., pn) such that pi+1 = 2pi + 1 for all 1 ≤ i < n. (Hence each term of such a chain except the last 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 pi+1 = 2pi − 1 for all 1 ≤ i < n.
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 pi+1 = api + b for all 1 ≤ i ≤ n for fixed coprime integers a and 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 and the next terms in the chain are not prime numbers.
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 = 7×17, but that is not prime.)
- 41, 83, 167 (The next number would be 335, but that is not prime.)
- 89, 179, 359, 719, 1439, 2879 (The next number would be 5759 = 13×443, 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 = 112, 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.
There are computing competitions for the longest Cunningham chain or for the one built up of the largest primes, but unlike the breakthrough of Ben J. Green and Terence Tao – the Green–Tao theorem, that there are arithmetic progressions of primes of arbitrary length – there is no general result known on large Cunningham chains to date.
|k||Kind||p1 (starting prime)||Digits||Year||Discoverer|
|1||1st / 2nd||282589933 − 1||24862048||2020||Curtis Cooper, GIMPS|
|2||1st||2618163402417×21290000 − 1||388342||2016||PrimeGrid|
|2nd||7775705415×2175115 + 1||52725||2017||Serge Batalov|
|3||1st||1815615642825×244044 − 1||13271||2016||Serge Batalov|
|2nd||742478255901×240067 + 1||12074||2016||Michael Angel & Dirk Augustin|
|4||1st||13720852541×7877# − 1||3384||2016||Michael Angel & Dirk Augustin|
|2nd||17285145467×6977# + 1||3005||2016||Michael Angel & Dirk Augustin|
|5||1st||31017701152691334912×4091# − 1||1765||2016||Andrey Balyakin|
|2nd||181439827616655015936×4673# + 1||2018||2016||Andrey Balyakin|
|6||1st||2799873605326×2371# - 1||1016||2015||Serge Batalov|
|2nd||52992297065385779421184×1531# + 1||668||2015||Andrey Balyakin|
|7||1st||82466536397303904×1171# − 1||509||2016||Andrey Balyakin|
|2nd||25802590081726373888×1033# + 1||453||2015||Andrey Balyakin|
|8||1st||89628063633698570895360×593# − 1||265||2015||Andrey Balyakin|
|2nd||2373007846680317952×761# + 1||337||2016||Andrey Balyakin|
|9||1st||553374939996823808×593# − 1||260||2016||Andrey Balyakin|
|2nd||173129832252242394185728×401# + 1||187||2015||Andrey Balyakin|
|10||1st||3696772637099483023015936×311# − 1||150||2016||Andrey Balyakin|
|2nd||2044300700000658875613184×311# + 1||150||2016||Andrey Balyakin|
|11||1st||73853903764168979088206401473739410396455001112581722569026969860983656346568919×151# − 1||140||2013||Primecoin (block 95569)|
|2nd||341841671431409652891648×311# + 1||149||2016||Andrey Balyakin|
|12||1st||288320466650346626888267818984974462085357412586437032687304004479168536445314040×83# − 1||113||2014||Primecoin (block 558800)|
|2nd||906644189971753846618980352×233# + 1||121||2015||Andrey Balyakin|
|13||1st||106680560818292299253267832484567360951928953599522278361651385665522443588804123392×61# − 1||107||2014||Primecoin (block 368051)|
|2nd||38249410745534076442242419351233801191635692835712219264661912943040353398995076864×47# + 1||101||2014||Primecoin (block 539977)|
|14||1st||4631673892190914134588763508558377441004250662630975370524984655678678526944768×47# − 1||97||2018||Primecoin (block 2659167)|
|2nd||5819411283298069803200936040662511327268486153212216998535044251830806354124236416×47# + 1||100||2014||Primecoin (block 547276)|
|15||1st||14354792166345299956567113728×43# - 1||45||2016||Andrey Balyakin|
|2nd||67040002730422542592×53# + 1||40||2016||Andrey Balyakin|
|2nd||2×1540797425367761006138858881 − 1||28||2014||Chermoni & Wroblewski|
|2nd||1540797425367761006138858881||28||2014||Chermoni & Wroblewski|
|18||2nd||658189097608811942204322721||27||2014||Chermoni & Wroblewski|
|19||2nd||79910197721667870187016101||26||2014||Chermoni & Wroblewski|
q# denotes the primorial 2 × 3 × 5 × 7 × ... × q.
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 base "shifts" the digits to the left; e.g. in decimal we have 314 × 10 = 3140.) 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.
- Primecoin, which uses Cunningham chains as a proof-of-work system
- Bi-twin chain
- Primes in arithmetic progression
- Joe Buhler, Algorithmic Number Theory: Third International Symposium, ANTS-III. New York: Springer (1998): 290
- Dirk Augustin, Cunningham Chain records. Retrieved on 2018-06-08.
- 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.
- The Prime Glossary: Cunningham chain
- Primecoin discoveries (primes.zone): online database of primecoin findings with list of records and visualization
- PrimeLinks++: Cunningham chain
- OEIS sequence A005602 (Smallest prime beginning a complete Cunningham chain of length n (of the first kind)) -- the first term of the lowest complete Cunningham chains of the first kind of length n, for 1 ≤ n ≤ 14
- OEIS sequence A005603 (Chains of length n of nearly doubled primes) -- the first term of the lowest complete Cunningham chains of the second kind with length n, for 1 ≤ n ≤ 15