Cunningham chain

In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes.

Definition

A Cunningham chain of the first kind of length n is a sequence of prime numbers (p₁, ..., p_n) such that p_i+1 = 2p_i + 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

{\begin{aligned}p_{2}&=2p_{1}+1,\\p_{3}&=4p_{1}+3,\\p_{4}&=8p_{1}+7,\\&{}\ \vdots \\p_{i}&=2^{i-1}p_{1}+(2^{i-1}-1),\end{aligned}}

or, by setting $a={\frac {p_{1}+1}{2}}$ (the number $a$ is not part of the sequence and need not be a prime number), we have $p_{i}=2^{i}a-1.$

Similarly, a Cunningham chain of the second kind of length n is a sequence of prime numbers (p₁, ..., p_n) such that p_i+1 = 2p_i − 1 for all 1 ≤ i < n.

It follows that the general term is

p_{i}=2^{i-1}p_{1}-(2^{i-1}-1).

Now, by setting $a={\frac {p_{1}-1}{2}}$ , we have $p_{i}=2^{i}a+1$ .

Cunningham chains are also sometimes generalized to sequences of prime numbers (p₁, ..., p_n) such that p_i+1 = ap_i + 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

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.)

89, 179, 359, 719, 1439, 2879 (The next number would be 5759, 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."^[1]

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.

Largest known Cunningham chain of length k (as of 17 March 2023^[2])
k	Kind	p₁ (starting prime)	Digits	Year	Discoverer
1	1st / 2nd	2^82589933 − 1	24862048	2018	Patrick Laroche, GIMPS
2	1st	2618163402417×2^1290000 − 1	388342	2016	PrimeGrid
2	2nd	213778324725×2⁵⁶¹⁴¹⁷ + 1	169015	2023	Ryan Propper & Serge Batalov
3	1st	1128330746865×2⁶⁶⁴³⁹ − 1	20013	2020	Michael Paridon
3	2nd	742478255901×2⁴⁰⁰⁶⁷ + 1	12074	2016	Michael Angel & Dirk Augustin
4	1st	13720852541×7877# − 1	3384	2016	Michael Angel & Dirk Augustin
4	2nd	49325406476×9811# + 1	4234	2019	Oscar Östlin
5	1st	31017701152691334912×4091# − 1	1765	2016	Andrey Balyakin
5	2nd	181439827616655015936×4673# + 1	2018	2016	Andrey Balyakin
6	1st	2799873605326×2371# - 1	1016	2015	Serge Batalov
6	2nd	52992297065385779421184×1531# + 1	668	2015	Andrey Balyakin
7	1st	82466536397303904×1171# − 1	509	2016	Andrey Balyakin
7	2nd	25802590081726373888×1033# + 1	453	2015	Andrey Balyakin
8	1st	89628063633698570895360×593# − 1	265	2015	Andrey Balyakin
8	2nd	2373007846680317952×761# + 1	337	2016	Andrey Balyakin
9	1st	553374939996823808×593# − 1	260	2016	Andrey Balyakin
9	2nd	173129832252242394185728×401# + 1	187	2015	Andrey Balyakin
10	1st	3696772637099483023015936×311# − 1	150	2016	Andrey Balyakin
10	2nd	2044300700000658875613184×311# + 1	150	2016	Andrey Balyakin
11	1st	73853903764168979088206401473739410396455001112581722569026969860983656346568919×151# − 1	140	2013	Primecoin (block 95569)
11	2nd	341841671431409652891648×311# + 1	149	2016	Andrey Balyakin
12	1st	288320466650346626888267818984974462085357412586437032687304004479168536445314040×83# − 1	113	2014	Primecoin (block 558800)
12	2nd	906644189971753846618980352×233# + 1	121	2015	Andrey Balyakin
13	1st	106680560818292299253267832484567360951928953599522278361651385665522443588804123392×61# − 1	107	2014	Primecoin (block 368051)
13	2nd	38249410745534076442242419351233801191635692835712219264661912943040353398995076864×47# + 1	101	2014	Primecoin (block 539977)
14	1st	4631673892190914134588763508558377441004250662630975370524984655678678526944768×47# − 1	97	2018	Primecoin (block 2659167)
14	2nd	5819411283298069803200936040662511327268486153212216998535044251830806354124236416×47# + 1	100	2014	Primecoin (block 547276)
15	1st	14354792166345299956567113728×43# - 1	45	2016	Andrey Balyakin
15	2nd	67040002730422542592×53# + 1	40	2016	Andrey Balyakin
16	1st	91304653283578934559359	23	2008	Jaroslaw Wroblewski
16	2nd	2×1540797425367761006138858881 − 1	28	2014	Chermoni & Wroblewski
17	1st	2759832934171386593519	22	2008	Jaroslaw Wroblewski
17	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.

As of 2018^[update], the longest known Cunningham chain of either kind is of length 19, discovered by Jaroslaw Wroblewski in 2014.^[2]

Congruences of Cunningham chains

Let the odd prime $p_{1}$ be the first prime of a Cunningham chain of the first kind. The first prime is odd, thus $p_{1}\equiv 1{\pmod {2}}$ . Since each successive prime in the chain is $p_{i+1}=2p_{i}+1$ it follows that $p_{i}\equiv 2^{i}-1{\pmod {2^{i}}}$ . Thus, $p_{2}\equiv 3{\pmod {4}}$ , $p_{3}\equiv 7{\pmod {8}}$ , 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 $p_{i+1}=2p_{i}+1$ in base 2, we see that, by multiplying $p_{i}$ by 2, the least significant digit of $p_{i}$ becomes the secondmost least significant digit of $p_{i+1}$ . Because $p_{i}$ is odd—that is, the least significant digit is 1 in base 2–we know that the secondmost least significant digit of $p_{i+1}$ is also 1. And, finally, we can see that $p_{i+1}$ will be odd due to the addition of 1 to $2p_{i}$ . 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:

Binary	Decimal
1000011011010000000100111101	141361469
10000110110100000001001111011	282722939
100001101101000000010011110111	565445879
1000011011010000000100111101111	1130891759
10000110110100000001001111011111	2261783519
100001101101000000010011110111111	4523567039

A similar result holds for Cunningham chains of the second kind. From the observation that $p_{1}\equiv 1{\pmod {2}}$ and the relation $p_{i+1}=2p_{i}-1$ it follows that $p_{i}\equiv 1{\pmod {2^{i}}}$ . In binary notation, the primes in a Cunningham chain of the second kind end with a pattern "0...01", where, for each $i$ , the number of zeros in the pattern for $p_{i+1}$ is one more than the number of zeros for $p_{i}$ . As with Cunningham chains of the first kind, the bits left of the pattern shift left by one position with each successive prime.

Similarly, because $p_{i}=2^{i-1}p_{1}+(2^{i-1}-1)\,$ it follows that $p_{i}\equiv 2^{i-1}-1{\pmod {p_{1}}}$ . But, by Fermat's little theorem, $2^{p_{1}-1}\equiv 1{\pmod {p_{1}}}$ , so $p_{1}$ divides $p_{p_{1}}$ (i.e. with $i=p_{1}$ ). Thus, no Cunningham chain can be of infinite length.^[3]

References

^ Joe Buhler, Algorithmic Number Theory: Third International Symposium, ANTS-III. New York: Springer (1998): 290
^ ^a ^b Norman Luhn & 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.

External links

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

[1] Joe Buhler, Algorithmic Number Theory: Third International Symposium, ANTS-III. New York: Springer (1998): 290

[records-2] Norman Luhn & Dirk Augustin, Cunningham Chain records. Retrieved on 2018-06-08.

[3] 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.

[1]

[2]

[3]