Consider prime `p`. In which cases are both `2*p+1` and `2*p-1` prime?

* Either `p = 3`, or `p` is indivisible by 3, which means `2 * p` is indivisible by 3.
* `2*p` is not divisible by 3. Either `2*p-1` or `2*p+1` must therefore be either exactly `3` or divisible by 3 (consecutive integers `2*p-1`, `2*p`, `2*p+1`)
 * In the latter case, the set `{p, 2*p±1}` contains exactly one element divisible by 3: that element is nonprime, meaning `p` does not start two valid chains.

The only possibilities remaining are if `p = 3` or `2*p±1 = 3`.

* If `p = 3`, `p` is part of `2`'s **type II** chain, and so does not start the chain; not a valid `p`.
* If `2*p+1 = 3`, then `p = 1` = a nonprime and so not a valid `p`.

The only remaining possible solution is `2*p-1 = 3`, where `p = 2`; the example already provided in the question.