I've written about or mentioned Cunningham chains in the following posts over the years:
Monday, 5 October 2015
Wednesday, 27 April 2016
Tuesday, 12 April 2016
Monday, 14 November 2016
Saturday, 30 April 2016
However, it's been over three years since that last past so it high time to say a little more about them. If, when a prime is multiplied by 2 and 1 added, a new prime is generated then the result is a Cunningham chain of the first type. If, when a prime is multiplied by 2 and 1 subtracted, a new prime is generated then the result is a Cunningham chain of the second type. The process is continued until a composite number is reached and the chain ends, having attained a certain length.
Today I turned 25645 days old and it so happened that this number is a member of OEIS A263311: numbers \(n\) such that each of \(p=6*n+1\), \(q=6*p+1\), \(r=6*q+1\) and \(s=6*r+1\) is prime. The first member of this sequence is 10 which yields \(p=61\), \(q=367\), \(r=2203\), \(s=13219\) and \(t=79315\). Thus we have the chain of primes 61, 367, 2203 and 13219 connected by the rule multiply by 6 and add 1. This is an example of a generalised Cunningham chain where, instead of multiplying by 2 and adding 1, we multiply by 6 and add 1. So to generalise even more, the primes in a generalised Cunningham chain are connected by the rule \(p_i=a*p+b\) where \(p\) is the first prime in the chain and \(a\) and \(b\) are coprime integers. So in the case of 25645, \(a=6\) and \(b=1\) and the primes thus generated are 147871, 887227, 5323363 and 31940179.
Here is another example. Let's consider the starting prime 5333 and the rule \(p_i=5*p+4\). This generates the sequence of primes 5333, 26669, 133349, 666749, 3333749, 16668749 and 83343749, representing a generalised Cunningham chain of length 7. As another example, if we start with 3203 and use the rule \(p_i=3*p+4\), we get the chain 3203, 9613, 28843, 86533 and 259603. Here are some more examples taken from this site. Figure 1 shows \(p_i=4*p-3\) with \(L\) being the number of primes in the chain with starting primes for the chain shown in the right column.
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| Figure 1 |
Figure 2 shows chains created using \(p_i=6*p-5\) and \(p_i=9*p-8\).
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| Figure 2 |
I found a paper (link) that discusses a connection between arithmetic derivatives and Cunningham chains. For example, the following is proposed:
A Cunningham chain of length 17 exists if and only if there exists a prime number \(p\) such that \(n = 2^4p\) satisfies the following differential equation:$$n^{17} = 2^{17} \cdot n + 524272$$
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