Smallest Cunningham Chain of the First Kind of Length 6

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Theorem

The smallest Cunningham chain of the first kind of length $6$ is:

$\tuple {89, 179, 359, 719, 1439, 2879}$


Proof

By definition, a Cunningham chain of the first kind is a sequence of prime numbers $\tuple {p_1, p_2, \ldots, p_n}$ such that:

$p_{k + 1} = 2 p_k + 1$
$\dfrac {p_1 - 1} 2$ is not prime
$2 p_n + 1$ is not prime.

Thus each term except the last is a Sophie Germain prime.


The sequence of Sophie Germain primes begins:

$2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, \ldots$


Let $P: \Z \to \Z$ be the mapping defined as:

$\map P n = 2 n + 1$


Applying $P$ iteratively to each of the smallest Sophie Germain primes in turn:


\(\ds \map P 2\) \(=\) \(\ds 5\)
\(\ds \map P 5\) \(=\) \(\ds 11\)
\(\ds \map P {11}\) \(=\) \(\ds 23\)
\(\ds \map P {23}\) \(=\) \(\ds 47\)
\(\ds \map P {47}\) \(=\) \(\ds 95\) which is not prime

Thus $\tuple {2, 5, 11, 23, 47}$ is a Cunningham chain of the first kind of length $5$.


\(\ds \map P 3\) \(=\) \(\ds 7\)
\(\ds \map P 7\) \(=\) \(\ds 15\) which is not prime


\(\ds \map P {29}\) \(=\) \(\ds 59\)
\(\ds \map P {59}\) \(=\) \(\ds 119\) which is not prime


\(\ds \map P {41}\) \(=\) \(\ds 83\)
\(\ds \map P {83}\) \(=\) \(\ds 167\)
\(\ds \map P {167}\) \(=\) \(\ds 335\) which is not prime


\(\ds \map P {53}\) \(=\) \(\ds 107\)
\(\ds \map P {107}\) \(=\) \(\ds 215\) which is not prime


\(\ds \map P {83}\) \(=\) \(\ds 167\)
\(\ds \map P {167}\) \(=\) \(\ds 335\) which is not prime


\(\ds \map P {89}\) \(=\) \(\ds 179\)
\(\ds \map P {179}\) \(=\) \(\ds 359\)
\(\ds \map P {359}\) \(=\) \(\ds 719\)
\(\ds \map P {719}\) \(=\) \(\ds 1439\)
\(\ds \map P {1439}\) \(=\) \(\ds 2879\)
\(\ds \map P {2879}\) \(=\) \(\ds 5759\) which is not prime.

It is noted that $\dfrac {89 - 1} 2 = 44$ which is not prime.

Hence the sequence of $6$:

$\tuple {89, 179, 359, 719, 1439, 2879}$

$\blacksquare$


Sources

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