A while ago, in another place, Peter Shaw observed that
there is a simple quadratic Ansatz that will generate BLS-provable
triplets. It's trivial to extend this to quads.
I just found a cubic Ansatz for almost-BLS quins.
By almost, I mean only a handful of digits short of BLS,
so one can always use KP and normally a bit of ECM will
avoid even this and give you BLS, thus obviating
APRCL or ECPP.
Now there might appear to be no point to this
observation, since quins are always going to be small
enough to make APRCL or ECPP very cheap.
But in fact there is a really nice point: if you
do a deep almost-BLS cubic-Ansatz five-fold sieve,
as if you were going to hunt for
quins, with all five channels almost-Carmody enriched,
but then only search for PrPs in the N+1 channel,
you then have, at negligible extra cost, 3 ways of getting
a triplet, and in all 3 cases any result is provable in no
time at all. Moreover you get a chance of a CPAP3 with d=6.
(Yes, Jim, I know that's not the best way for CPAP3,
but no harm to add one almost-free if statement to the
triply-enhanced triplet search.)
I might have a go at a 3k-digit Primo-free triplet, this way,
one fine day. But not real soon.