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Re^5: check for square-number with a regex

by JadeNB (Chaplain)
on Nov 26, 2009 at 23:35 UTC ( #809669=note: print w/replies, xml ) Need Help??

in reply to Re^4: check for square-number with a regex
in thread check for square-number with a regex

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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