## Prime Sieves Using Binary Quadratic Forms (1999)

Venue: | Mathematics of Computation |

Citations: | 10 - 1 self |

### BibTeX

@ARTICLE{Atkin99primesieves,

author = {A. O. L. Atkin and D. J. Bernstein},

title = {Prime Sieves Using Binary Quadratic Forms},

journal = {Mathematics of Computation},

year = {1999},

volume = {73},

pages = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We introduce an algorithm that computes the prime numbers up to N using O(N=log log N) additions and N 1=2+o(1) bits of memory. The algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms. 1.

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A linear algorithm for incremental digital display of circular arcs
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Citation Context ...xsx + 10. Go back to step 2. PRIME SIEVES USING BINARY QUADRATIC FORMS 5 Notes. Tracing a level curve is a standard technique in computer graphics; see, e.g., [1, chapter 17]. It is often credited to =-=[4-=-] but it appeared earlier in [8, section 3]. 5. Asymptotic performance For large N one can compute the primes up to N as follows. Dene W as 12 times the product of all the primes from 5 up to about p ... |

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Citation Context ... bits of fast memory. Notes. Singleton in [13] suggested chopping a large interval into small pieces and applying the sieve of Eratosthenes to each piece. The same idea was published independently in =-=[3]-=- and later in [2]. Sieving an arithmetic progression is the p-adic analogue of sieving a bounded interval. Presumably Eratosthenes did not bother writing down even numbers in his sieve. Instead of run... |

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Explaining the wheel sieve
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(Show Context)
Citation Context ...ry by the method of section 2. By a similar method one can compute the primes up to N using O(N=log log N) operations and N 1+o(1) bits of memory. Pritchard gave a proof in [9] and a simpler proof in =-=[10]-=-. Dunten, Jones, and Sorenson in [5] reduced the amount of memory by a factor of log N . The new method is simultaneously within a constant factor of the best known number of operations and within N o... |

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Fast compact prime number sieves (among others
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(Show Context)
Citation Context ...hm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms. 1. Introduction Pritchard in =-=[11] as-=-ked whether it is possible to print the prime numbers up to N , in order, using o(N) operations and O(N ) bits of memory for some s1. Here \memory" does not include the paper used by the printer... |

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(Show Context)
Citation Context ...ory. Notes. Singleton in [13] suggested chopping a large interval into small pieces and applying the sieve of Eratosthenes to each piece. The same idea was published independently in [3] and later in =-=[2]-=-. Sieving an arithmetic progression is the p-adic analogue of sieving a bounded interval. Presumably Eratosthenes did not bother writing down even numbers in his sieve. Instead of running Algorithm 2.... |

7 | Elementary Number Theory, McGrawHill - Uspensky, Heaslet - 1939 |

2 | Experiments on the lattice problem - Keller, Swenson |

1 |
A space-ecient fast prime number sieve
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(Show Context)
Citation Context ...imilar method one can compute the primes up to N using O(N=log log N) operations and N 1+o(1) bits of memory. Pritchard gave a proof in [9] and a simpler proof in [10]. Dunten, Jones, and Sorenson in =-=[5]-=- reduced the amount of memory by a factor of log N . The new method is simultaneously within a constant factor of the best known number of operations and within N o(1) of the best known amount of memo... |

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A sublinear additive sieve for prime numbers
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(Show Context)
Citation Context ...Algorithm 2.1 independently for each d, one can handle all d simultaneously for each q:snd all nontrivial multiples of q between 30L and 30L + 30B, and translate each multiple into a pair (k; d). See =-=[9]-=- for details. For suciently large q this saves time despite the added cost of translation. One can include composite integers q in step 2 of Algorithm 2.1. For example, it is easy to run through all i... |

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1 |
Algorithm 357: an ecient prime number generator
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(Show Context)
Citation Context ...using the gcc 2.8.1 compiler on an UltraSPARC-I/167, takes 19:6 seconds tosnd the 50847534 primes up to 1000000000. Here B = 128128; the UltraSPARC has 131072 bits of fast memory. Notes. Singleton in =-=[13]-=- suggested chopping a large interval into small pieces and applying the sieve of Eratosthenes to each piece. The same idea was published independently in [3] and later in [2]. Sieving an arithmetic pr... |