## Sharpening PRIMES is in P for a large family of numbers (2005)

Venue: | Math. Comp |

Citations: | 9 - 0 self |

### BibTeX

@ARTICLE{Berrizbeitia05sharpeningprimes,

author = {Pedro Berrizbeitia},

title = {Sharpening PRIMES is in P for a large family of numbers},

journal = {Math. Comp},

year = {2005},

pages = {2006--11191}

}

### OpenURL

### Abstract

We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algorithms n n we present run in 2 − min(k,[2 log log n]) Õ(log n) 6 time, where k = ν2(n − 1) is the exact power of 2 dividing n − 1 when n ≡ 1 (mod 4) and k = ν2(n + 1) if n ≡ −1 (mod 4). The complexity of our algorithms improves up to Õ(log n)4 when k ≥ [2 log log n]. We also give tests for more general family of numbers and study their complexity.

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Citation Context ...rovided there is a prime q, q ≡ 1(modp), such that n is not a p-th power modulo q, and gave many concrete implementations and tables of primes. Further extensions of Williams’ results can be found in =-=[7]-=-. The book of Williams [18] is a good source for studying many of these results and the history of this subject. Polynomial time deterministic primality tests for larger families have also been studie... |

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Citation Context ..., q ≡ 1(modp), such that n is not a p-th power modulo q, and gave many concrete implementations and tables of primes. Further extensions of Williams’ results can be found in [7]. The book of Williams =-=[18]-=- is a good source for studying many of these results and the history of this subject. Polynomial time deterministic primality tests for larger families have also been studied. For instance, Konyagin a... |