Re: Prime Number Finder

[gu]

There's an interesting algorithm for finding whether a number is prime in the documentation of Quantum::Superpositions, which provides quantum-like superpositions in Perl :

use Quantum::Superpositions ;

sub is_prime {
    my ($n) = @_;
    return $n % all(2..sqrt($n)+1) != 0
}

See the documentation for details on how quantum-like superpositions work.

Gu

Update : Removed assertion on the complexity of this algorithm

[blokhead]

Nice succinct algorithm, but I must take issue with

There's an interesting O(1) algorithm

You do have to execute the algorithm on a classical computer, so Q::S or not, it's most definitely not O(1). It'll be exponential (in the number of bits in $n) because behind the scenes, Q::S is dividing $n by all possible factors (what else could it be doing?). But even on a quantum computer, you still need either a division or gcd circuit (and probably a lot of other stuff), which will take some polynomial time in the number of bits.

Just because it's a one-liner doesn't make it O(1). Anyway, my favorite cutesy inefficient primality checker is

sub is_prime {
   ("1" x $_[0]) !~ /^(11+)\1+$/
}

[Roy Johnson]

some polynomial time in the number of bits.

If the number of bits is constant, then any polynomial time based on it is constant.

[blokhead]

If the number of bits is constant, then any polynomial time based on it is constant.

Big-O statements (like an algorithm taking constant or O(1) time) are statements about asymptotic behavior, i.e, how the function behaves in the limit (usually, as input size tends to infinity). If you don't look at them in the limit, then big-O-ish language (like constant time) is meaningless.

How meaningless? Even undecidable languages have a constant time "algorithm" if you consider the input size to be held to a constant. So without viewing things in the limit, all problems become computationally equivalent in the asymptotic language.

Update: added citation from parent node

[hv]

Even if you had a quantum computer to run it on, I'm not convinced you'd have an O(1) algorithm unless you also have O(1) algorithms for computing sqrt($n) and 2..$n.

In any case however the algorithm will give the wrong answer for 2. You can fix that by replacing sqrt($n)+1 with sqrt($n+1).

Hugo