C#: How to make Sieve of Atkin incremental

Question

I don\'t know if this is possible or not, but I just gotta ask. My mathematical and algorithmic skills are kind of failing me here :P

The thing is I now have this cl

Answer 1

I can try to explain what x and y does, but I don't think you can do what you ask without restarting the loops from the beginning. This is pretty much the same for any "sieve"-algorithm.

What the sieve does is basically count how many different quadratic equations (even or odd) have each number as a solution. The specific equation checked for each number is different depending on what n % 12 is.

For example, numbers n which have a mod 12 remainder of 1 or 5 are prime if and only if the number of solutions to 4*x^2+y^2=n is odd and the number is square-free. The first loop simply loops through all possible values of x and y that could satisfy these different equations. By flipping the isPrime[n] each time we find a solution for that n, we can keep track of whether the number of solutions is odd or even.

The thing is that we count this for all possible n at the same time, which makes this much more efficient than checking one at the time. Doing it for just some n would take more time (because you would have to make sure that n >= lower_limit in the first loop) and complicate the second loop, since that one requires knowledge of all primes smaller than sqrt.

The second loop checks that the number is square-free (has no factor which is a square of a prime number).

Also, I don't think your implementation of the sieve of Atkin is necessarily faster than a straight-forward sieve of Eratosthenes would be. However, the fastest possible most optimized sieve of Atkin would beat the fastest possible most optimized sieve of Eratosthenes.