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Primes In Elliptic Divisibility Sequences
"... 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 vie ..."
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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.
Primes In Divisibility Sequences
"... 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. ..."
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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.
On the primality of F 4723 and F 5387
, 1999
"... Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the nth 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)], ..."
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Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the nth 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
Some Facts for LucasLehmer Primality Testers
"... . 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; ..."
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. 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]. ...
Irish Math. Soc. Bulletin 54 (2004), 63–72 63 Note on the Lucas–Lehmer Test
"... Abstract. A proof of the Lucas–Lehmer test can be difficult to find, for most textbooks that state the result do not prove it. Over the past two decades, there have been some efforts to produce elementary versions of this famous result. However, the two that we acknowledge in this note did so by usi ..."
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Abstract. A proof of the Lucas–Lehmer test can be difficult to find, for most textbooks that state the result do not prove it. Over the past two decades, there have been some efforts to produce elementary versions of this famous result. However, the two that we acknowledge in this note did so by using either algebraic numbers or group theory. It also appears that in the process of trying to develop an elementary proof of this theorem, the original version provided by D. H. Lehmer’s in 1930 has been overlooked. Furthermore, it is quite succinct and elementary, although by its style, it appears to have been written for an audience that has expertise in the theory of the extended Lucas sequences. Therefore, it is the primary objective of this paper to provide a brief introduction into the theory that underlies the said sequences, as well as to present an annotated version of Lehmer’s original proof of the Lucas–Lehmer test. In conclusion, we show how the test may be utilized in order to identify certain composite
NOTE ON FIBONACCI PRIMALITY TESTING
, 1996
"... One of the most effective ways of proving an integer N is prime is to show first that N is a probable prime, i.e., that a N ~ l = 1 (mod N) for some base a and l<a<Nl, and then to find enough prime factors of N±l so that certain other conditions are satisfied (see [1] for details of such prim ..."
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One of the most effective ways of proving an integer N is prime is to show first that N is a probable prime, i.e., that a N ~ l = 1 (mod N) for some base a and l<a<Nl, and then to find enough prime factors of N±l so that certain other conditions are satisfied (see [1] for details of such primality tests). The problem of finding these prime factors is, of course, the difficult and
Identities for the Common Factors of Fibonacci and Lucas numbers
"... In this paper, we obtain the identities for the common factors of Fibonacci and Lucas numbers. New identities for even and odd Fibonacci numbers are obtained. ..."
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In this paper, we obtain the identities for the common factors of Fibonacci and Lucas numbers. New identities for even and odd Fibonacci numbers are obtained.