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Re^2: To Findout Prime number

by ZlR (Chaplain)
on Jul 17, 2009 at 12:31 UTC ( #781043=note: print w/replies, xml ) Need Help??


in reply to Re: To Findout Prime number
in thread To Findout Prime number

perl -wle 'print "Prime" if (1 x shift) !~ /^1?$|^(11+?)\1+$/'

OK so I thought i'd elaborate a little on how this works since the monks from the chatterbox took time to explain it to me :)

So you have a number, say 6, and "write" it as a sequence of 1 ie (111111) .
If you write it (11)(11)(11) "you just found that 6 = 3 * 2", as moritz explained it.

ELISHEVA explains the math behind this :
if there is a repeating group of the same number of ones, then a factorization is possible and hence the number can't be prime.

It means that if you can write the number as M groups of K ones, then it factorizes as :

M * [ Sum(p=1->k) 1 ] which really is just M * K

Which means that our number can be divided by M (or K), and therefore it's not prime. Shmem gives an example: i.e. for m = 1763, the group found would be 11111111111111111111111111111111111111111 repeated 43 times - not prime

So how does the regexp implement that ? We can decompose it like this :

m/ ^ # start of line 1? # the number 1, zero or one time $ # end of line | # alternation ^ # start of line ( # remember the match in \1 11+? # the number 1, then the number 1 once or more t +imes, but the less time possible ) # \1+ # the matched sequence, once or more. $ /x ;
So the real trick is in (11+?). In a standalone context, it will only match 11. That's because +? means "once or more but the minimum number of times". On the other hand, ^(11+?)$ will match a whole sequence of ones from begining to end.

Here (11+?) is followed by \1+$ which means : itself, once or more, until the end of the line .
So what happens is that the "minimum number of times" that's contained in +? is seen from the \1+$ that is after it in the expression
So when no match occurs, the engine will go back to the (11+?) and try again.

I understand that's what backtracking is. Eventually the regexp engine will try every grouping of ones and fine none.
Since it needs at least two groupings to match (enforced by the + in \1+ ), a prime number will not match.

The only problem is that for some numbers you get a Complex regular subexpression recursion limit (32766) exceeded error. My guess was that it happens when the engine has to try more than 32766 number of times, ie the first fail appears for a number that needs K>32766. That's not the case though, since prime number 32779 does not yield the error. 65558 does, though. I went on and brutforced it, to find that the error appears first with 65536, which is 2 * 32768, which computes since it needs two groups to match...

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[ambrus]: Corion: yes, you need to implement the io watcher, which should be simple because Prima::File is basically that, and the timer watcher form Prima::Timer
[Corion]: ... or so I think. As I said, I'm somewhat vague on how to make AnyEvent cooperate with a callback-driven IO event loop that gives me callbacks when data is available or can be written
[ambrus]: what push_write thing? I don't think you need that. that's implemented generically by AnyEvent::Handle
[Corion]: ambrus: Yeah, that's what I think as well. But you give me an idea, maybe I should start with implementing the timer, as that should be far simpler and with fewer edge-cases/nasty interaction than the file watcher
[ambrus]: You only provide the watcher part that tells when the handle is readable or writable, not the actual writing and reading.
[Corion]: ambrus: Hmmm. It makes sense that AnyEvent would implement the push_write itself, but I think I don't have a good idea of where the boundary between AnyEvent and the underlying event system lies... Implementing the timer should give me a better idea
[ambrus]: Corion: push_write is in the higher level abstraction of AnyEvent::Handle, not in the watcher
[Corion]: ambrus: Hmm - rereading Prima::File, that merrily coincides with what Prima does - it tells you "you can read", and you're supposed to read from the fh yourself. I thought it called you with the data already read, which would've been harder to integrate
[ambrus]: you just need an io watcher, created by &AnyEvent::Impl:: Whatever::io(...)
[Corion]: So after talking it through with you even while I'm still not entirely clear on where AE ends and my implementation begins, I think I understand that I only need to implement some smaller parts for each functionality I want to support.

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