Smallest Cunningham Chain of the First Kind of Length 12

Theorem
The smallest Cunningham chain of length $12$ is:
 * $554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$,

Proof
Let $C$ denote the sequence in question.

We have 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, as is required 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.