A faster pseudo-primality test

Faster Primality Testing (Extended Abstract)

Graph Ambiguity and Pseudo-primality

Recognising the types of graph on which certain distributed algorithms may fail is NP-complete. One of the results can be restated as the NP-completeness of deciding whether a graph is a non-trivial covering of some (unknown) other graph. R esum e Nous consid erons deux familles de graphes intervenant dans des prob-l emes d'algorithmique distribu ee: la famille de graphes ambigus et la famille ...

A Potentially Fast Primality Test

The running time is O(r log n). It can be shown by elementary means that the required r exists in O(log n). So the running time is O(log n). Moreover, by Fouvry’s Theorem [8], such r exists in O(log n), so the running time becomes O(log n). In [10], Lenstra and Pomerance showed that the AKS primality test can be improved by replacing the polynomial x − 1 in equation (1.1) with a specially const...

Miller's Primality Test

Primality test, prime In this paper we prove the following simplification of Miller's primality criterion [2]. Theorem 1. Assume that for every integer d that is 1 mod 4 and either prime or the product of two primes, the L-function I= & (k/d) l kmS satisfies the generalized Riemann hypothesis, where {kid) denotes the Jacobi symbol, defined below. Let n be an odd integer, n > I , and write n-I-=...

Biquadratic reciprocity and a Lucasian primality test

Let {sk, k ≥ 0} be the sequence defined from a given initial value, the seed, s0, by the recurrence sk+1 = s 2 k − 2, k ≥ 0. Then, for a suitable seed s0, the number Mh,n = h · 2n − 1 (where h < 2n is odd) is prime iff sn−2 ≡ 0 mod Mh,n. In general s0 depends both on h and on n. We describe a slight modification of this test which determines primality of numbers h·2n±1 with a seed which depends...