Smallest Cunningham Chain of the First Kind of Length 7

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Theorem

The smallest Cunningham chain of the first kind of length $7$ is:

$\left({1 \, 122 \, 659, 2 \, 245 \, 319, 4 \, 490 \, 639, 8 \, 981 \, 279, 17 \, 962 \, 559, 35 \, 925 \, 119, 71 \, 850 \, 239}\right)$


Proof

Let $C$ denote the sequence in question.

We have that:

$\dfrac {1 \, 122 \, 659 - 1} 2 = 561 \, 329 = 83 \times 6763$

and so is not prime.

Thus $1 \, 122 \, 659$ is not a safe prime, as is required for $C$ to be a Cunningham chain of the first kind.


Then:

\(\ds 2 \times 561 \, 329 + 1\) \(=\) \(\ds 1 \, 122 \, 659\) which is the $87 \, 359$th prime
\(\ds 2 \times 1 \, 122 \, 659 + 1\) \(=\) \(\ds 2 \, 245 \, 319\) which is the $165 \, 760$th prime
\(\ds 2 \times 2 \, 245 \, 319 + 1\) \(=\) \(\ds 4 \, 490 \, 639\) which is the $315 \, 347$th prime
\(\ds 2 \times 4 \, 490 \, 639 + 1\) \(=\) \(\ds 8 \, 981 \, 279\) which is the $601 \, 286$th prime
\(\ds 2 \times 8 \, 981 \, 279 + 1\) \(=\) \(\ds 17 \, 962 \, 559\) which is the $1 \, 149 \, 096$th prime
\(\ds 2 \times 17 \, 962 \, 559 + 1\) \(=\) \(\ds 35 \, 925 \, 119\) which is the $2 \, 199 \, 933$rd prime
\(\ds 2 \times 35 \, 925 \, 119 + 1\) \(=\) \(\ds 71 \, 850 \, 239\) which is the $4 \, 220 \, 407$th prime
\(\ds 2 \times 71 \, 850 \, 239 + 1\) \(=\) \(\ds 143 \, 700 \, 479\)
\(\ds \) \(=\) \(\ds 13 \times 47 \times 479 \times 491\) and so is not prime


Establishing that this is indeed the smallest such Cunningham chain of the first kind of length $7$ can be done by a computer search.

$\blacksquare$


Sources

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