E. Fouvry. Theoreme de Brun-Titchmarsh; application au theoreme de Fermat. Invent. Math., 79:383-407, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is evaluated by repeated squaring) or even better O(r log p) if Fast Fourier Multiplication [Knu98] is used. Our algorithm first chooses a suitable r. An r is suitable for us if it is a prime=O(log p) and r 1 contains a prime factor of size at least r 2 # , for some constant # 0. [Fou85, BH96] assures us that such a suitable r exists. Thereafter, the algorithm verifies the congruence (2) for a small ( O( # r log p) number of a s. We prove that this idea works: i.e. the algorithm correctly determines whether p is prime or not. 3 Notation and Preliminaries This section states ....

....we have 1. Thus the order of x in this field must be r (since r is prime and x 1) Therefore r (p 1) i.e. 1 (mod r) Hence, d k. Therefore, k = d, and the lemma follows. In addition to the above algebraic facts, we will need the following two number theoretic facts. Lemma 3.2. [Fou85, BH96] Let P (n) denote the greatest prime divisor of n. There exist constants c 0 and n 0 such that, for all x n 0 p p is prime, p x and P (p 1) x # c log x . The above lemma is, in fact, known to hold for exponents upto 0.6683 (see [BH96] for a summary of results of this kind. ....

E. Fouvry. Theoreme de Brun-Titchmarsh; application au theoreme de Fermat. Invent. Math., 79:383--407, 1985.

No context found.

E. Fouvry. Theoreme de Brun-Titchmarsh; application au theoreme de Fermat. Invent. Math., 79:383-407, 1985.

No context found.

E. Fouvry. Theoreme de Brun-Titchmarsh; application au theoreme de Fermat. Invent. Math., 79:383--407, 1985.

No context found.

E. Fouvry. Theoreme de Brun-Titchmarsh; application au theoreme de Fermat. Invent. Math., 79:383--407, 1985.

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