## Improved Incremental Prime Number Sieves (1994)

Venue: | Cornell University |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Pritchard94improvedincremental,

author = {Paul Pritchard},

title = {Improved Incremental Prime Number Sieves},

booktitle = {Cornell University},

year = {1994},

pages = {280--288},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental: -- it employs only \Theta(n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for fixed n; -- the transition from n to n + 1 takes only O(1) additions of numbers of size O(n). (On average, of course, O(1) such additions increase the limit up to which all primes are known from n to n + \Theta(log log n)). 1 Introduction A so-called "formula" for the i'th prime has been a long-lived concern, if not quite the Holy Grail, of Elementary Number Theory. This concern seems poorly motivated, as evidenced by the extraordinary freak-show of solutions proffered over the ages. The natural setting is Algorithmic Number Theory, and what is desired is much better cast as an algorithm to compute the i'th prime. Given that app...

### Citations

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The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...ratosthenes' sieve was an improvement on most of its successors. In this connection, note that the asymptotically fastest known multiplication algorithm for a RAM, that of Schonhage and Strassen (see =-=[2]-=-), has a complexity equal to that of \Theta(log log N log log log N ) additions. So the bit-complexity of Eratosthenes' sieve is lower than one of \Theta(N ) multiplications. The latter is characteris... |

466 |
The Science of Programming
- Gries
- 1981
(Show Context)
Citation Context ... k : the i'th greatest member of W k ; W k : (fgx : 1sxsx mod \Pi k 2 W k ); g k : (Max x : x 2 W k : next(W k ; x) \Gamma x). Our algorithms are expressed in the language of guarded commands used in =-=[4]-=-, extended with if- and forall-commands. The command if b then S fi is an abbreviation of if b ! S :b ! skip fi: The forall-command denotes iteration over a fixed finite set in unspecified order (see ... |

295 |
An Introduction to the Theory
- Hardy, Wright
- 1978
(Show Context)
Citation Context ...ltiplication table. In the case that f is the greatest member \Pi k \Gamma 1, the next number is p k+2 \Delta (\Pi k + 1), which differs from n by (p k+2 \Gamma p k+1 ) \Delta \Pi k + p k+2 + p k+1 : =-=(5)-=- Unfortunately, \Pi k falls outside the bounds of the multiplication table. However, the product in (5) may be incrementally computed by repeated additions of \Pi k . Since p k+2 \Gamma p k+1 = O(d pk... |

34 |
A note on the differences between consecutive primes
- Goldston, Heath-Brown
- 1984
(Show Context)
Citation Context ... p 0 , so p ! p n and hence p 0 ! 2 p n by Bertrand's Theorem. Therefore p 0 \Gamma p ! d 2 p n : The best known upper bound on prime gaps is the very conservative dn = O(n 0:55+ffl ) for any ffl ? 0 =-=[6]-=-. Thus p 0 \Gamma p = O(n 0:275+ffl ) for any ffl ? 0: Now consider the second multiplicand. lastf (p i ) and f are either p i and p i+1 respectively, or successive members of W i+1 . In the former ca... |

26 | Open problems in number-theoretic complexity ii
- Adleman, McCurley
- 1994
(Show Context)
Citation Context ...O(n 0:275+ffl ) by O(n 0:432 ) multiplication table suffices for the product (p 0 \Gamma p) \Delta (f \Gamma lastf (p)). In passing, we note that this is certainly a gross over-estimate. According to =-=[1], Cram'er's conjectu-=-re that dn = \Theta(log 2 n) "has been called into question", but the slightly weaker dn = O(log 2+ffl n) for any ffl ? 0 is "still probably true". Suppose this latter claim is tru... |

17 |
A sublinear additive sieve for finding prime numbers
- Pritchard
- 1981
(Show Context)
Citation Context ...er is characteristic of many proposed parallel algorithms (let alone some sequential ones). Nevertheless, progress was eventually made. The fastest known (sequential) algorithm is now our wheel sieve =-=[9, 10]-=-, which requires \Theta(N= log log N ) additions. It enjoys the properties of being sublinear, i.e., o(N ), and additive, i.e., not requiring multiplications. Bengalloun [3] promoted another desiderat... |

9 |
On the problem of Jacobsthal
- Iwaniec
- 1978
(Show Context)
Citation Context ...pectively, or successive members of W i+1 . In the former case, f \Gamma lastf (p i ) = O(n 0:275+ffl ) as above. In the latter, f \Gamma lastf (p i ) = O(g i+1 ) = O(p 2 i+1 ) by a result of Iwaniec =-=[7]-=-. Since f = n \Xi p i , the difference is also bounded by dn\Xip i = O((n \Xi p i ) 0:55+ffl ): We may approximately balance these two known bounds as follows. If p i = O(n 0:216 ), the difference is ... |

9 |
Fast compact prime number sieves (among others
- Pritchard
- 1983
(Show Context)
Citation Context ... numbers not divisible by one of the first k primes. g k is the maximum gap between successive natural numbers not divisible by one of the first k primes. We use the following notation (introduced in =-=[11]-=-): ;: the empty set; jSj: the cardinality of set S; next(S; x): (Min y : y 2 Ssy ? x); a::b: (fgx : asxsb); (x; y): the g.c.d. of x and y; x j y: x divides y; p i : the i'th prime number; P rimes(S): ... |

9 |
Opportunistic algorithms for eliminating supersets
- Pritchard
- 1991
(Show Context)
Citation Context ... a consistent notation, due to E. W. Dijkstra, wherever a variable is bound by a quantifier such as 8, 9, P , Q , Max, Min and fg. The set-forming quantifier fg is our invention; it was first used in =-=[13]-=-. The general form for a set-constructor is (fgx : D(x) : t(x)) which denotes the set of values t(x) when x ranges over the domain characterized by D(x). However, t(x) is commonly x, and in such cases... |

7 |
Explaining the wheel sieve
- Pritchard
- 1982
(Show Context)
Citation Context ...er is characteristic of many proposed parallel algorithms (let alone some sequential ones). Nevertheless, progress was eventually made. The fastest known (sequential) algorithm is now our wheel sieve =-=[9, 10]-=-, which requires \Theta(N= log log N ) additions. It enjoys the properties of being sublinear, i.e., o(N ), and additive, i.e., not requiring multiplications. Bengalloun [3] promoted another desiderat... |

1 |
An incremental primal sieve
- Bengalloun
- 1986
(Show Context)
Citation Context ...is now our wheel sieve [9, 10], which requires \Theta(N= log log N ) additions. It enjoys the properties of being sublinear, i.e., o(N ), and additive, i.e., not requiring multiplications. Bengalloun =-=[3]-=- promoted another desideratum, that of being incremental. This means that N need not be fixed, so that the algorithm can in theory be run indefinitely to enumerate the primes. Bengalloun's incremental... |

1 |
On the prime example of programming
- Pritchard
- 1980
(Show Context)
Citation Context ...o N may be stored in O(N log log N= log N ) bits and still recovered in order in O(1) additive operations per prime. Our new algorithm lacks one important property: it is not compact. As we showed in =-=[8, 11]-=-, Eratosthenes' sieve and some linear wheel-based sieves can be implemented to run in O(N 0:5 ) bits, whereas our new algorithm and Bengalloun 's sieve require O(N log N ) bits (to enumerate the prime... |

1 |
Linear prime number sieves: a family tree
- Pritchard
- 1987
(Show Context)
Citation Context ..., extended with if- and forall-commands. The command if b then S fi is an abbreviation of if b ! S :b ! skip fi: The forall-command denotes iteration over a fixed finite set in unspecified order (see =-=[12]-=-). 3 Bengalloun's Sieve The following discussion of Bengalloun's sieve recapitulates our presentation in [12]. Bengalloun's basic sieve is based on the following normal form for composites c: c = p \D... |