Prime Divisors of Sequences Associated to Elliptic Curves

Everest, Graham; Shparlinski, Igor E.

doi:10.1017/s0017089504002113

Cited by 7 publications

(6 citation statements)

References 23 publications

(33 reference statements)

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“…Somos 4 sequences, and EDS in particular, are of considerable interest to number theorists due to the way that primitive prime divisors appear [6,7,9,10,11,12,35]. In that and in other respects, they have a lot in common with linear recurrence sequences (see [8, chapters 10 and 11]).…”

Section: Introductionmentioning

confidence: 99%

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Hone

Swart

2008

Math. Proc. Camb. Phil. Soc.

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Somos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

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Section: Introductionmentioning

confidence: 99%

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Hone

Swart

2008

Math. Proc. Camb. Phil. Soc.

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“…In the introduction of [3], it is claimed that there exists a preprint by Everest and King with a proof of our Theorem 1.9. This preprint was never published and we were not able to find the preprint.…”

Section: Remark 111mentioning

confidence: 98%

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Verzobio

2021

Res. number theory

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Let P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

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“…Remark 1.9. In the introduction of [2], it is claimed that there exists a preprint by Everest and King with a proof of our Theorem 1.7. This preprint was never published and we were not able to find the preprint.…”

Section: Introductionmentioning

confidence: 98%

“…2 . This follow easily from the study of the complex curve E ′ (C) and in particular from definition of ℘ E ′ (z) (see[9, Chapter VI]), since this function is close to z −2 for z close to 0.…”

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confidence: 99%

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Verzobio

2020

Preprint

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Consider P and Q two points on an elliptic curve defined over a number field M . For α ∈ End(E), define Bα as the denominator of x(α(P ) + Q), which is an integral ideal in M . Let O be a subring of End(E) and we will study the sequence {Bα}α∈O. We will show that, for all but finitely many α ∈ O, Bα has a primitive divisor, assuming that O is a Dedekind domain, that P is a non-torsion point, and that there exist two endomorphisms g = 0 and f so that f (P ) = g(Q).

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Prime Divisors of Sequences Associated to Elliptic Curves

Cited by 7 publications

References 23 publications

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Primitive divisors of sequences associated to elliptic curves with complex multiplication

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