Sieve of Eratosthenes

In mathematics, the sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e. not prime) the multiples of each prime, starting with the multiples of 2.^[1]

The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them which is equal to that prime.^[1] This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.^[2]

The sieve of Eratosthenes is one of the most efficient ways to find all of the smaller primes (below 10 million or so).^[3] It is named after Eratosthenes of Cyrene, a Greek mathematician; although none of his works has survived, the sieve was described and attributed to Eratosthenes in the Introduction to Arithmetic by Nicomachus.^[4]

The sieve may be used to find primes in arithmetic progressions.^[5]

Algorithm description

A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself.

To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method:

1. Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n).
2. Initially, let p equal 2, the first prime number.
3. Starting from p, enumerate its multiples by counting to n in increments of p, and mark them in the list (these will be 2p, 3p, 4p, etc.; the p itself should not be marked).
4. Find the first number greater than p in the list that is not marked. If there was no such number, stop. Otherwise, let p now equal this new number (which is the next prime), and repeat from step 3.

When the algorithm terminates, all the numbers in the list that are not marked are prime.

The main idea here is that every value for p is prime, because we have already marked all the multiples of the numbers less than p. Note that some of the numbers being marked may have already been marked earlier (e.g. 15 will be marked both for 3 and 5).

As a refinement, it is sufficient to mark the numbers in step 3 starting from p², as all the smaller multiples of p will have already been marked at that point. This means that the algorithm is allowed to terminate in step 4 when p² is greater than n.^[1]

Another refinement is to initially list odd numbers only, (3, 5, ..., n), and count in increments of 2p in step 3, thus marking only odd multiples of p. This actually appears in the original algorithm.^[1] This can be generalized with wheel factorization, forming the initial list only from numbers coprime with the first few primes and not just from odds, i.e. numbers coprime with 2.^[9]

Incremental sieve

An incremental formulation of the sieve^[2] generates primes indefinitely (i.e. without an upper bound) by interleaving the generation of primes with the generation of their multiples (so that primes can be found in gaps between the multiples), where the multiples of each prime p are generated directly, by counting up from the square of the prime in increments of p (or 2p for odd primes). The generation must be initiated only when the prime's square is reached, to avoid adverse effects on efficiency.

Trial division

Trial division can be used to produce primes by filtering out the composites found by testing each candidate number for divisibility by its preceding primes. It is often confused with the sieve of Eratosthenes, although the latter directly generates the composites instead of testing for them. Trial division has worse theoretical complexity than that of the sieve of Eratosthenes in generating ranges of primes.^[2]

When testing each candidate number, the optimal trial division algorithm uses just those prime numbers not exceeding its square root. The widely known 1975 functional code by David Turner^[10] is often presented as an example of the sieve of Eratosthenes^[9] but is actually a sub-optimal trial division algorithm.^[2]

Example

To find all the prime numbers less than or equal to 30, proceed as follows.

First generate a list of integers from 2 to 30:

 2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

First number in the list is 2; cross out every 2nd number in the list after it (by counting up in increments of 2), i.e. all the multiples of 2:

 2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Next number in the list after 2 is 3; cross out every 3rd number in the list after it (by counting up in increments of 3), i.e. all the multiples of 3:

 2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Next number not yet crossed out in the list after 3 is 5; cross out every 5th number in the list after it (by counting up in increments of 5), i.e. all the multiples of 5:

 2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Next number not yet crossed out in the list after 5 is 7; the next step would be to cross out every 7th number in the list after it, but they are all already crossed out at this point, as these numbers (14, 21, 28) are also multiples of smaller primes because 7*7 is greater than 30. The numbers left not crossed out in the list at this point are all the prime numbers below 30:

 2  3     5     7           11    13          17    19          23                29

Algorithm complexity

Time complexity of calculating all primes below n in the random access machine model is operations, a direct consequence of the fact that the prime harmonic series asymptotically approaches . It has an exponential time complexity with regard to input size, though, which makes it a pseudo-polynomial algorithm.

The bit complexity of the algorithm is bit operations with a memory requirement of .^[11]

The segmented version of the sieve of Eratosthenes, with basic optimizations, uses operations and bits of memory.^[12]

Implementation

An optimized implementation (starting from squares), in pseudocode:^[13][14]

Input: an integer n > 1
 
Let A be an array of Boolean values, indexed by integers 2 to n,
initially all set to true.
 
 for i = 2, 3, 4, ..., not exceeding √n:
  if A[i] is true:
    for j = i2, i2+i, i2+2i, ..., not exceeding n :
      A[j] := false

The result of the algorithm is an array A[] where A[i] is true if and only if i is prime.

Large ranges may not fit entirely in memory. In these cases it is necessary to use a segmented sieve where only portions of the range are sieved at a time.^[15]

For ranges with upper limit n so large that the sieving primes below √n as required by the sieve of Eratosthenes cannot fit in memory, a slower but much more space-efficient sieve like that of Sorenson can be used instead.^[16]

Euler's Sieve

Euler's proof of the zeta product formula contains a version of the sieve of Eratosthenes in which each composite number is eliminated exactly once.^[14] It, too, starts with a list of numbers from 2 to n in order. On each step the first element is identified as the next prime and the results of multiplying this prime with each element of the list are marked in the list for subsequent deletion. The initial element and the marked elements are then removed from the working sequence, and the process is repeated:

[2] (3) 5  7  9  11  13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79  ...
[3]    (5) 7     11  13    17 19    23 25    29 31    35 37    41 43    47 49    53 55    59 61    65 67    71 73    77 79  ...
[4]       (7)    11  13    17 19    23       29 31       37    41 43    47 49    53       59 61       67    71 73    77 79  ...
[5]             (11) 13    17 19    23       29 31       37    41 43    47       53       59 61       67    71 73       79  ...
[...]

Here the example is shown starting from odds, after the first step of the algorithm. Thus, on the kth step all the remaining multiples of the kth prime are removed from the list, which will thereafter contain only numbers coprime with the first k primes (cf., wheel factorization), so that the list will start with the next prime, and all the numbers in it below the square of its first element will be prime too.

Thus, when generating a bounded sequence of primes, when the next identified prime exceeds the square root of the upper limit, all the remaining numbers in the list are prime.^[14] In the example given above that is achieved on identifying 11 as next prime, giving a list of all primes less than or equal to 80.

Note that numbers that will be discarded by a step are still used while marking the multiples in that step, e.g., for the multiples of 3 it is 3 · 3 = 9, 3 · 5 = 15, 3 · 7 = 21, 3 · 9 = 27, ..., 3 · 15 = 45, ... , so care must be taken dealing with this.^[14]

See also

• Sieve of Atkin
• Sieve of Sundaram
• Sieve theory

References

External links

• Eratosthenes, sieve of at Encyclopaedia of Mathematics
• Interactive JavaScript Page
• Sieve of Eratosthenes by George Beck, Wolfram Demonstrations Project.
• Sieve of Eratosthenes in Haskell
• Sieve of Eratosthenes algorithm illustrated and explained. Java and C++ implementations.
• A related sieve written in x86 assembly language
• A highly optimized Sieve of Eratosthenes in C
• A parallel implementation in C#
• SieveOfEratosthenesInManyProgrammingLanguages c2 wiki page
• The Art of Prime Sieving Sieve of Eratosthenes in C from 1998 with nice features and algorithmic tricks explained.