## DETERMINISTIC METHODS TO FIND PRIMES

### BibTeX

@MISC{Polymath_deterministicmethods,

author = {D. H. J. Polymath},

title = {DETERMINISTIC METHODS TO FIND PRIMES},

year = {}

}

### OpenURL

### Abstract

Abstract. Given a large positive integer N, how quickly can one construct a prime number larger than N (or between N and 2N)? Using probabilistic methods, one can obtain a prime number in time at most log O(1) N with high probability by selecting numbers between N and 2N at random and testing each one in turn for primality until a prime is discovered. However, if one seeks a deterministic method, then the problem is much more difficult, unless one assumes some unproven conjectures in number theory; brute force methods give a O(N 1+o(1) ) algorithm, and the best unconditional algorithm, due to Odlyzko, has a run time of O(N 1/2+o(1)). In this paper we discuss an approach that may improve upon the O(N 1/2+o(1)) bound, by suggesting a strategy to determine in time O(N 1/2−c) for some c> 0 whether a given interval in [N, 2N] contains a prime. While this strategy has not been fully implemented, it can be used to establish partial results, such as being able to determine the parity of the number of primes in a given interval in [N, 2N] in time O(N 1/2−c). 1.

### Citations

70 |
and J.Pintz, The difference between consecutive primes
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(Show Context)
Citation Context ...(1) |E|) units of work. Thus, for instance, using Bertrand’s postulate one recovers the O(N 1+o(1) ) bound; using the unconditional fact that [N, N + N 0.525 ] contains a prime for every large N (see =-=[2]-=-) we improve this to O(N 0.525+o(1) ); and if one assumes the Riemann hypothesis, then as is well known we obtain a prime in an interval of the form [N, N + N 0.5+o(1) ] for all large N, leading to a ... |

69 | Elements of Number Theory - Vinogradov |

44 |
On the order of magnitude of the difference between consecutive prime numbers
- Cramér
- 1936
(Show Context)
Citation Context ...m that runs in time O(N 1/k+o(1) ). Unfortunately the asymptotic for primes of the form n k + 1 is not known even for k = 2, which is a famous open conjecture of Landau. A famous conjecture of Cramér =-=[3]-=- (see also [5] for refinements) asserts that the largest prime gap in [N, 2N] is of the order of O(log 2 N), which would give a deterministic algorithm with run time O(log O(1) N). Unfortunately, this... |

21 |
Schnelle Multiplikation grosser Zahlen, Computing 7
- Schönhage
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(Show Context)
Citation Context ... )0≤i,j<Q; B := (t V (i,j) )0≤j,k<Q; C := (t W (k,i) )0≤k,i<Q. Each of the matrices A, B, C has a circuit complexity of O(N o(1) Q 2 ). Using the SchönhageStrassen fast matrix multiplication algorithm=-=[10]-=-, one can multiply A, B, C together using a circuit of complexity O(Q 3−c0 ) for some absolute constant c0 > 0. Taking the trace requires another circuit of complexity O(Q). Putting all these circuits... |

19 | Harald Cramér and the distribution of prime numbers
- Granville
- 1995
(Show Context)
Citation Context ... time O(N 1/k+o(1) ). Unfortunately the asymptotic for primes of the form n k + 1 is not known even for k = 2, which is a famous open conjecture of Landau. A famous conjecture of Cramér [3] (see also =-=[5]-=- for refinements) asserts that the largest prime gap in [N, 2N] is of the order of O(log 2 N), which would give a deterministic algorithm with run time O(log O(1) N). Unfortunately, this conjecture is... |

14 |
Computing π(x): the Meissel-Lehmer method
- Lagarias, Miller, et al.
- 1985
(Show Context)
Citation Context ...s less than or equal to x, for x in [N, 2N], since an interval [a, b] contains a prime if and only if π(b)−π(a−1) > 0. The fastest known elementary method to compute π(x) is the Meissel-Lehmer method =-=[7]-=-, [4], which takes time O(x 2/3 / log 2 x) and leads to a O(N 2/3+o(1) ) algorithm. On the other hand, if one can calculate π(x) for x ∈ [N, 2N] approximately using A units of work to a guaranteed err... |

12 |
Primes represented by x 3 +2y 3
- Heath-Brown
(Show Context)
Citation Context ...val of the form [N, N + N 0.5+o(1) ] for all large N, leading to a bound of O(N 0.5+o(1) ). There are other sparse sets that are known to contain primes. For instance, using the result of Heath-Brown =-=[6]-=- that there are infinitely many primes of the form a 3 + 2b 3 (which comes with the expected asymptotic), the above strategy gives an unconditional algorithm with time O(N 2/3+o(1) ), since the number... |

11 |
Computing π(x): the Meissel
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(Show Context)
Citation Context ...s than or equal to x, for x in [N, 2N], since an interval [a, b] contains a prime if and only if π(b)−π(a−1) > 0. The fastest known elementary method to compute π(x) is the Meissel-Lehmer method [7], =-=[4]-=-, which takes time O(x 2/3 / log 2 x) and leads to a O(N 2/3+o(1) ) algorithm. On the other hand, if one can calculate π(x) for x ∈ [N, 2N] approximately using A units of work to a guaranteed error of... |

9 | Computing π(x): An analytic method
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- 1987
(Show Context)
Citation Context ...be careful of is to ensure in the binary search algorithm is that the density of primes in the interval is always ≫ 1/ log N, but this is easily accomplished.) It was observed by Lagarias and Odlyzko =-=[8]-=- that by using an explicit contour integral formula for π(x) (or the closely related expression ψ(x) = ∑ n≤x Λ(n)) in terms of the Riemann zeta function, that one could compute π(x) to accuracy L usin... |