February 12, 2010
We examined the use of the Sieve of Eratosthenes for enumerating the prime numbers, a method that dates to the ancient Greeks, in two previous exercises. Just a few years ago, in 2004, A. O. L. Atkin and D. J. Bernstein developed a new method, called the Sieve of Atkin, that performs the same task more quickly. The new method first marks squarefree potential primes, then eliminates those potential primes that are not squarefree.
Atkin’s sieve begins with a boolean array (bits are fine) of length n equal to the number of items to be sieved; each element of the array is initially false.
Squarefree potential primes are marked like this: For each element k of the array for which k ≡ 1 (mod 12) or k ≡ 5 (mod 12) and there exists some 4x² + y² = k for positive integers x and y, flip the sieve element (set true elements to false, and false elements to true). For each element k of the array for which k ≡ 7 (mod 12) and there exists some 3x² + y² = k for positive integers x and y, flip the sieve element. For each element k of the array for which k ≡ 11 (mod 12) and there exists some 3x² – y² = k for positive integers x and y with x > y, flip the sieve element.
Once this preprocessing is complete, the actual sieving is performed by running through the sieve starting at 7. For each true element of the array, mark all multiples of the square of the element as false (regardless of their current setting). The remaining true elements are prime.
Your task is to write a function to sieve primes using the Sieve of Atkin. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.