Smallest Cunningham Chain of the First Kind of Length 12

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Theorem

The smallest Cunningham chain of the first kind of length $12$ is:

$554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$, $2 \, 218 \, 753 \, 113 \, 719$, $4 \, 437 \, 506 \, 227 \, 439$,
$8 \, 875 \, 012 \, 454 \, 879$, $17 \, 750 \, 024 \, 909 \, 759$, $35 \, 500 \, 049 \, 819 \, 519$, $71 \, 000 \, 099 \, 639 \, 039$,
$142 \, 000 \, 199 \, 278 \, 079$, $284 \, 000 \, 398 \, 556 \, 159$, $568 \, 000 \, 797 \, 112 \, 319$, $1 \, 136 \, 001 \, 594 \, 224 \, 639$


Proof

Let $C$ denote the sequence in question.

We have that $554 \, 688 \, 278 \, 429$ is prime.


First note that:

$\dfrac {554 \, 688 \, 278 \, 429 - 1} 2 = 277 \, 344 \, 139 \, 214 = 2 \times 138 \, 672 \, 069 \, 607$

and so is not prime.

Thus $554 \, 688 \, 278 \, 429$ is not a safe prime, and thus fulfils the requirement for $C$ to be a Cunningham chain of the first kind.


Then:

\(\text {(1)}: \quad\) \(\ds 2 \times 554 \, 688 \, 278 \, 429 + 1\) \(=\) \(\ds 1 \, 109 \, 376 \, 556 \, 859\) which is prime
\(\text {(2)}: \quad\) \(\ds 2 \times 1 \, 109 \, 376 \, 556 \, 859 + 1\) \(=\) \(\ds 2 \, 218 \, 753 \, 113 \, 719\) which is prime
\(\text {(3)}: \quad\) \(\ds 2 \times 2 \, 218 \, 753 \, 113 \, 719 + 1\) \(=\) \(\ds 4 \, 437 \, 506 \, 227 \, 439\) which is prime
\(\text {(4)}: \quad\) \(\ds 2 \times 4 \, 437 \, 506 \, 227 \, 439 + 1\) \(=\) \(\ds 8 \, 875 \, 012 \, 454 \, 879\) which is prime
\(\text {(5)}: \quad\) \(\ds 2 \times 8 \, 875 \, 012 \, 454 \, 879 + 1\) \(=\) \(\ds 17 \, 750 \, 024 \, 909 \, 759\) which is prime
\(\text {(6)}: \quad\) \(\ds 2 \times 17 \, 750 \, 024 \, 909 \, 759 + 1\) \(=\) \(\ds 35 \, 500 \, 049 \, 819 \, 519\) which is prime
\(\text {(7)}: \quad\) \(\ds 2 \times 35 \, 500 \, 049 \, 819 \, 519 + 1\) \(=\) \(\ds 71 \, 000 \, 099 \, 639 \, 039\) which is prime
\(\text {(8)}: \quad\) \(\ds 2 \times 71 \, 000 \, 099 \, 639 \, 039 + 1\) \(=\) \(\ds 142 \, 000 \, 199 \, 278 \, 079\) which is prime
\(\text {(9)}: \quad\) \(\ds 2 \times 142 \, 000 \, 199 \, 278 \, 079 + 1\) \(=\) \(\ds 284 \, 000 \, 398 \, 556 \, 159\) which is prime
\(\text {(10)}: \quad\) \(\ds 2 \times 284 \, 000 \, 398 \, 556 \, 159 + 1\) \(=\) \(\ds 568 \, 000 \, 797 \, 112 \, 319\) which is prime
\(\text {(11)}: \quad\) \(\ds 2 \times 568 \, 000 \, 797 \, 112 \, 319 + 1\) \(=\) \(\ds 1 \, 136 \, 001 \, 594 \, 224 \, 639\) which is prime
\(\text {(12)}: \quad\) \(\ds 2 \times 1 \, 136 \, 001 \, 594 \, 224 \, 639 + 1\) \(=\) \(\ds 2 \, 272 \, 003 \, 188 \, 449 \, 279\) which is $19 \times 119 \, 579 \, 115 \, 181 \, 541$ prime


Establishing that this is indeed the smallest such Cunningham chain of the first kind of length $12$ can be done by a computer search.

$\blacksquare$


Sources

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