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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 = 2 pi + 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 p_2 = 2p_1+1, p_3 = 4p_1+3, p_4 = 8p_1+7, ..., p_i = 2^{i-1}p_1 + (2^{i-1}-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 = 2 pi - 1.

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.

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.

Largest known Cunningham chain of length k (as of July 3 2008)
k Kind p1 (starting prime) Digits Year Discoverer
2 1st 48047305725×2172403 − 1 51910 2007 D. Underbakke
3 1st 164210699973×226326 − 1 7937 2006 M. Paridon
4 1st 119184698×5501# − 1 2354 2005 J. Sun
5 1st 357487161295×1693# − 1 727 2008 D. Augustin
6 2nd 37783362904×1097# + 1 475 2006 D. Augustin
7 1st 1178137903623×661# − 1 291 2008 D. Augustin
8 1st 2×65728407627×431# − 1 186 2005 D. Augustin
9 1st 65728407627×431# − 1 185 2005 D. Augustin
10 2nd 145683282311×181# + 1 84 2005 D. Augustin
11 2nd 2×(8428860×127# + 212148902055091) − 1 56 2006 J. K. Andersen
12 2nd 8428860×127# + 212148902055091 56 2006 J. K. Andersen
13 1st 1753286498051×71# − 1 39 2005 D. Augustin
14 2nd 2×28320350134887132315879689643841 − 1 32 2008 J. Wroblewski
15 2nd 28320350134887132315879689643841 32 2008 J. Wroblewski
16 2nd 2368823992523350998418445521 28 2008 J. Wroblewski
17 2nd 1302312696655394336638441 25 2008 J. Wroblewski

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

As of July 2008, 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 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.

References

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