Sieve of Atkin
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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.^{[1]}
Contents 
[edit] Algorithm
In the algorithm:
 All remainders are modulosixty 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.
 Create a results list, filled with 2, 3, and 5.
 Create a sieve list with an entry for each positive whole number; all entries of this list should initially be marked nonprime.
 For each entry in the sieve list :
 If the entry is for a number with remainder 1, 13, 17, 29, 37, 41, 49, or 53, flip it for each possible solution to 4x^{2} + y^{2} = entry_number.
 If the entry is for a number with remainder 7, 19, 31, or 43, flip it for each possible solution to 3x^{2} + y^{2} = entry_number.
 If the entry is for a number with remainder 11, 23, 47, or 59, flip it for each possible solution to 3x^{2}  y^{2} = entry_number when x > y.
 If the entry has some other remainder, ignore it completely.
 Start with the lowest number in the sieve list.
 Take the next number in the sieve list still marked prime.
 Include the number in the results list.
 Square the number and mark all multiples of that square as nonprime.
 Repeat steps five through eight.
[edit] 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) ∧ (n mod 12 = 1 ∨ n mod 12 = 5): is_prime(n) ← ¬is_prime(n) n ← 3x²+y² if (n ≤ limit) ∧ (n mod 12 = 7): is_prime(n) ← ¬is_prime(n) n ← 3x²y² if (x > y) ∧ (n ≤ limit) ∧ (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 squarefree 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].
[edit] Explanation
The algorithm completely ignores any numbers divisible by two, three, or five. All numbers with modulosixty 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 modulosixty remainder 3, 9, 15, 21, 27, 33, 39, 45, 51, or 57 are divisible by three and not prime. All numbers with modulosixty remainder 5, 25, 35, or 55 are divisible by five and not prime. All these remainders are ignored.
All numbers with modulosixty remainder 1, 13, 17, 29, 37, 41, 49, or 53 have a modulofour remainder of 1. These numbers are prime if and only if the number of solutions to 4x^{2} + y^{2} = n is odd and the number is squarefree (proven as theorem 6.1 of ^{[1]}).
All numbers with modulosixty remainder 7, 19, 31, or 43 have a modulosix remainder of 1. These numbers are prime if and only if the number of solutions to 3x^{2} + y^{2} = n is odd and the number is squarefree (proven as theorem 6.2 of ^{[1]}).
All numbers with modulosixty remainder 11, 23, 47, or 59 have a modulotwelve remainder of 11. These numbers are prime if and only if the number of solutions to 3x^{2} − y^{2} = n is odd and the number is squarefree (proven as theorem 6.3 of ^{[1]}).
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 2^{2}, 3^{2}, and 5^{2}.
[edit] 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^{[1]}. These asymptotic computational complexities include simple optimizations, such as wheel factorization, and splitting the computation to smaller blocks.
[edit] Addendum
The first two equations used to determine if a number is prime after their respective modulo tests are equations for ellipses. They can be rewritten into standard form for an ellipse by dividing both sides of the equation by n, where n is the entry number being tested for primality. Using the equations in this form is easier to implement a test for various reasons. See ellipse for more information.
[edit] References
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} A.O.L. Atkin, D.J. Bernstein, Prime sieves using binary quadratic forms, Math. Comp. 73 (2004), 10231030.[1]
[edit] External links
 An optimized implementation of the sieve (in C)
 Python implementation
 Simple implementation of the sieve (in C)
