Smallest Cunningham Chain of the First Kind of Length 12

Theorem
The smallest Cunningham chain 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.

Then:

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