Primitive divisors of elliptic divisibility sequences

Graham Everest; Gerard McLaren; Thomas Ward

doi:10.1016/j.jnt.2005.08.002

Primitive divisors of elliptic divisibility sequences

Graham Everest, Gerard McLaren, Thomas Ward

Research output: Contribution to journal › Article › peer-review

31 Citations (Scopus)

Abstract

{Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.

Original language	English
Pages (from-to)	71-89
Number of pages	19
Journal	Journal of Number Theory
Volume	118
Issue number	1
DOIs	https://doi.org/10.1016/j.jnt.2005.08.002
Publication status	Published - 2006

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Cite this

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title = "Primitive divisors of elliptic divisibility sequences",

abstract = "{Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.",

author = "Graham Everest and Gerard McLaren and Thomas Ward",

year = "2006",

doi = "10.1016/j.jnt.2005.08.002",

language = "English",

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journal = "Journal of Number Theory",

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publisher = "Academic Press Inc.",

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T1 - Primitive divisors of elliptic divisibility sequences

AU - Everest, Graham

AU - McLaren, Gerard

AU - Ward, Thomas

PY - 2006

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N2 - {Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.

AB - {Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.

U2 - 10.1016/j.jnt.2005.08.002

DO - 10.1016/j.jnt.2005.08.002

M3 - Article

VL - 118

SP - 71

EP - 89

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -