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.

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

\begin{align} 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{align}

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,...,pn) such that for all 1 ≤ i < n, pi+1 = 2pi − 1.

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, ..., pn) such that for all 1 ≤ i ≤ n, pi+1api + b for fixed coprime integers ab; 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."[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.

Largest known Cunningham chain of length k (as of 14 December 2013[2])
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
3 1st 914546877×234772 − 1 10477 2010 Tom Wu
4 1st 1249097877×6599# − 1 2835 2011 Michael Angel
5 1st 4250172704×2749# − 1 1183 2012 Dirk Augustin
6 1st 37488065464×1483# − 1 633 2010 Dirk Augustin
7 1st 162597166369×827# − 1 356 2010 Dirk Augustin
8 2nd 1148424905221×509# + 1 224 2010 Dirk Augustin
9 1st 65728407627×431# − 1 185 2005 Dirk Augustin
10 1st 44598464649019035883154084128331646888059795218766083584048621139159337786287845212160000×149# − 1 146 2013 Primecoin
11 1st 73853903764168979088206401473739410396455001112581722569026969860983656346568919×151# − 1 140 2013 Primecoin
12 1st 61592551716229060392971860549140211602858978086524024531871935735163762961673908480×71# − 1 110 2013 Primecoin
13 2nd 10756750720700195380397697188448178460115725467111771468875842964723844354555016704×31# + 1 94 2013 Primecoin
14 2nd 335898524600734221050749906451371 33 2008 Jens Kruse Andersen
15 2nd 28320350134887132315879689643841 32 2008 Jaroslaw Wroblewski
16 2nd 2368823992523350998418445521 28 2008 Jaroslaw Wroblewski
17 2nd 1302312696655394336638441 25 2008 Jaroslaw Wroblewski

q# denotes the primorial 2×3×5×7×...×q.

As of 2013, 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.[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 number of the base "shifts" the digits to the left.) 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 $p_{i+1} = 2 p_i - 1$ 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

1. ^ Joe Buhler, Algorithmic Number Theory: Third International Symposium, ANTS-III. New York: Springer (1998): 290
2. ^ a b Dirk Augustin, Cunningham Chain records. Retrieved on 2013-12-14.
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.