I also think that the regex primality test is outside of CFG, but I've never proven that, so don't rely on it ;-)
Doesn't the pumping lemma that you cited also show what you claim? Specifically, if there were a CFG that recognised exactly the strings of
1s of prime length, then we would have the assertion that any sufficiently large prime
s could be written as
s = u + v + x + y + z, with everyone non-negative and
v + y at least
1, in such a way that
u + x + i*(v + y) + z was also prime for every non-negative
i. However, in that case,
- if u + x + z = 0, then we can take i to be composite, so that u + x + z + i*(v + y) = i*(v + y) is composite;
or
- otherwise, we may take i = (v + y + 2)*(u + x + z), so that u + x + z + i*(v + y) = (v + y + 1)^2*(u + x + z) is composite.
This is a contradiction.
UPDATE: Forgot to handle the case u + x + z = 0.
UPDATE 2: Oops, and I was quite sloppy in my choice of i even when u + x + z is positive. I think that it's OK now.
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