Sieve of Atkin

In mathematics, the sieve of Atkin is a fast, modern algorithm for finding all prime numbers up to a specified integer. It is an optimized version of the ancient sieve of Eratosthenes, but does some preliminary work and then marks off multiples of primes squared, rather than multiples of primes. It was created by A. O. L. Atkin and Daniel J. Bernstein.

Algorithm

In the algorithm:

• All remainders are modulo-sixty remainders (divide the number by sixty and return the remainder).
• All numbers, including x and y, are whole numbers (positive integers).
• Flipping an entry in the sieve list means to change the marking (prime or nonprime) to the opposite marking.

1. Create a results list, filled with 2, 3, and 5.
2. Create a sieve list with an entry for each positive whole number; all entries of this list should initially be marked nonprime.
3. For each entry number n in the sieve list, with modulo-sixty remainder r :
   • If r is 1, 13, 17, 29, 37, 41, 49, or 53, flip the entry for each possible solution to 4x² + y² = n.
   • If r is 7, 19, 31, or 43, flip the entry for each possible solution to 3x² + y² = n.
   • If r is 11, 23, 47, or 59, flip the entry for each possible solution to 3x² − y² = n when x > y.
   • If r is something else, ignore it completely.
4. Start with the lowest number in the sieve list.
5. Take the next number in the sieve list still marked prime.
6. Include the number in the results list.
7. Square the number and mark all multiples of that square as nonprime.
8. Repeat steps five through eight.

• This results in numbers with an odd number of solutions to the corresponding equation being prime, and an even number being nonprime.

Pseudocode

The following is pseudocode for a straightforward version of the algorithm:

// arbitrary search limit
limit ← 1000000         

// initialize the sieve
is_prime(i) ← false, i ∈ [5, limit] 

// put in candidate primes: 
// integers which have an odd number of
// representations by certain quadratic forms
for (x, y) in [1, √limit] × [1, √limit]:
    n ← 4x²+y²
    if (n ≤ limit) and (n mod 12 = 1 or n mod 12 = 5):
        is_prime(n) ← ¬is_prime(n)
    n ← 3x²+y²
    if (n ≤ limit) and (n mod 12 = 7):
        is_prime(n) ← ¬is_prime(n)
    n ← 3x²-y²
    if (x > y) and (n ≤ limit) and (n mod 12 = 11):
        is_prime(n) ← ¬is_prime(n)
  
// eliminate composites by sieving
for n in [5, √limit]:
    if is_prime(n):
        // n is prime, omit multiples of its square; this is
        // sufficient because composites which managed to get
        // on the list cannot be square-free
        is_prime(k) ← false, k ∈ {n², 2n², 3n², ..., limit} 

print 2, 3
for n in [5, limit]:
    if is_prime(n): print n

This pseudocode is written for clarity. Repeated and wasteful calculations mean that it would run slower than the sieve of Eratosthenes. To improve its efficiency, faster methods must be used to find solutions to the three quadratics. At the least, separate loops could have tighter limits than [1, √limit].

Explanation

The algorithm completely ignores any numbers divisible by two, three, or five. All numbers with modulo-sixty remainder 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, or 58 are divisible by two and not prime. All numbers with modulo-sixty remainder 3, 9, 15, 21, 27, 33, 39, 45, 51, or 57 are divisible by three and not prime. All numbers with modulo-sixty remainder 5, 25, 35, or 55 are divisible by five and not prime. All these remainders are ignored.

All numbers with modulo-sixty remainder 1, 13, 17, 29, 37, 41, 49, or 53 have a modulo-four remainder of 1. These numbers are prime if and only if the number of solutions to 4x² + y² = n is odd and the number is squarefree (proven as theorem 6.1 of ).

All numbers with modulo-sixty remainder 7, 19, 31, or 43 have a modulo-six remainder of 1. These numbers are prime if and only if the number of solutions to 3x² + y² = n is odd and the number is squarefree (proven as theorem 6.2 of ).

All numbers with modulo-sixty remainder 11, 23, 47, or 59 have a modulo-twelve remainder of 11. These numbers are prime if and only if the number of solutions to 3x² − y² = n is odd and the number is squarefree (proven as theorem 6.3 of ).

None of the potential primes are divisible by 2, 3, or 5, so they can't be divisible by their squares. This is why squarefree checks don't include 2², 3², and 5².

Computational complexity

This sieve computes primes up to N using O(N/log log N) operations with only N^1/2 + o(1) bits of memory. That is a little better than the sieve of Eratosthenes which uses O(N) operations and O(N^1/2(log log N)/log N) bits of memory. These asymptotic computational complexities include simple optimizations, such as wheel factorization, and splitting the computation to smaller blocks.

External links

• An optimized implementation of the sieve (in C)
• Python implementation