Primes in Elliptic Divisibility Sequences

Einsiedler, Manfred; Everest, Graham; Ward, Thomas

doi:10.1112/s1461157000000772

Cited by 43 publications

(76 citation statements)

References 16 publications

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“…The arithmetical properties of EDS and Somos sequences -in particular the distribution of primes therein -are a subject of current interest [10,11,49]. Some of these properties are discussed in the book [12] (see section 1.1.20, for instance), where it is suggested that such bilinear recurrences should be suitable generalizations of linear ones, with many analogous features.…”

Section: Somos Sequences and The Laurent Propertymentioning

confidence: 99%

Laurent Polynomials and Superintegrable Maps

Hone¹

2007

SIGMA

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Abstract. This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations.

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Section: Somos Sequences and The Laurent Propertymentioning

confidence: 99%

Laurent Polynomials and Superintegrable Maps

Hone¹

2007

SIGMA

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“…Ward, who made an extensive study of these sequences [24,25], shows that a proper elliptic divisibility sequence W is associated to a (possibly singular) elliptic curve E W and point P W ∈ E W (Q) and that the values of W n are closely related to the values of the division polynomials F n (P W ). More recently, elliptic divisibility sequences have been studied by Shipsey [19], who gives an application to the elliptic curve discrete logarithm problem, and by several other authors [3,4,5,8,9,10,23,22]. (See also [11,13,14] for work on the related Somos sequences.)…”

Section: Theorem 3 With Notation and Assumptions As Inmentioning

confidence: 99%

“…It is easy to check that it suffices that (W n ) satisfy the two relations [24].) Additional material on elliptic divisibility sequences may be found in [4,5,8,9,10,19,23,22,24,25].…”

Section: Elliptic Divisibility Sequencesmentioning

confidence: 99%

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p-adic properties of division polynomials and elliptic divisibility sequences

Silverman

2004

Math. Ann.

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Abstract. For a fixed rational point P ∈ E(K) on an elliptic curve, we consider the sequence of values F n (P ) n≥1 of the division polynomials of E at P . For a finite field K/F p , we prove that the sequence is periodic. For a local field K/Q p , we prove (under certain hypotheses) that there is a power q = p e so that for all m ≥ 1, the limit of F mq k (P ) as k → ∞ exists in K and is algebraic over Q(E). We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.

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“…See for instance [EEW01] or [Rey12] and the references therein. In this paper we derive strong conditions on the denominators d P from the uniform abc-conjecture over number fields (see Conjecture 2.2 or [GS00]).…”

Section: Introductionmentioning

confidence: 99%