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

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

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

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

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

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

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.

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

A similar result holds for Cunningham chains of the second kind. From the observation that ${\displaystyle p_{1}\equiv 1{\pmod {2))}$ and the relation ${\displaystyle p_{i+1}=2p_{i}-1}$ it follows that ${\displaystyle p_{i}\equiv 1{\pmod {2^{i))))$. In binary notation, the primes in a Cunningham chain of the second kind end with a pattern "0...01", where, for each ${\displaystyle i}$, the number of zeros in the pattern for ${\displaystyle p_{i+1))$ is one more than the number of zeros for ${\displaystyle p_{i))$. 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 ${\displaystyle p_{i}=2^{i-1}p_{1}+(2^{i-1}-1)\,}$ it follows that ${\displaystyle p_{i}\equiv 2^{i-1}-1{\pmod {p_{1))))$. But, by Fermat's little theorem, ${\displaystyle 2^{p_{1}-1}\equiv 1{\pmod {p_{1))))$, so ${\displaystyle p_{1))$ divides ${\displaystyle p_{p_{1))}$ (i.e. with ${\displaystyle i=p_{1))$). Thus, no Cunningham chain can be of infinite length.^[3]