Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.

We give an overview of two important families of divisibility sequences: the Lehmer{Pierce family (which generalise the Mersenne sequence) and the elliptic divisibility sequences. Recent computational work is described, as well as some of the mathematics behind these sequences.

Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the n-th Fibonacci number (resp. Lucas number). The aim of this informal note is to describe a short proof of primality for both F 4723 and F 5387 . See the paper [5] for more on this topic. 2 F 4723 From [3, (4.1)], one has F 4k+3 \Gamma 1 = F k+1 L k+1 L 2k+1 : (1) Here k = 1180, k + 1 = 1181 and 2k + 1 = 2361 = 3 \Theta 787. From [3] and with the help of factors found by Montgomery and Silverman [7, 8, 9], we get F 1181 = 5453857 \Theta C 240 ; L 1181 = 59051 \Theta<F27.43

. Let n be a probable prime, such that n k \Gamma 1 = F \Delta R, with F ? 3 p n being totally factored. In generalized Lucas -- Lehmer primality tests [5], one builds a ring A oe ZZ=(n \Delta ZZ) of degree k and seeks elements ff with ff n k \Gamma1 = 1 and (a n k \Gamma1 q \Gamma 1; n) = 1; 8qjF: In this paper we survey answers to some common questions arising in the context of Lucas -- Lehmer proofs for large numbers. These are: 1. How to combine known factors of n \Gamma 1, n + 1 and further \Phi k (n), with \Phi k , the k\Gammath cyclotomic polynomial, for small k. 2. How to optimize the required number of exponentiations for a complete proof. 3. How many bases ff does one need to try in average. The answers we present are in some case improvements of known results, restrictions to special cases, or new evaluations. Together they may provide some useful general tools for Lucas -- Lehmer testers. The first question is answered in general by Lenstra in [5] and [6]. ...