Primitive divisors of elliptic divisibility sequences

Everest, G.; McLaren, G.; Ward, T.

doi:10.1016/j.jnt.2005.08.002

Primitive divisors of elliptic divisibility sequences

Everest, G.; McLaren, G.; Ward, T.

Authors

G. Everest

G. McLaren

T. Ward

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.

Citation

Everest, G., McLaren, G., & Ward, T. (2006). Primitive divisors of elliptic divisibility sequences. Journal of Number Theory, 118(1), 71-89. https://doi.org/10.1016/j.jnt.2005.08.002

Journal Article Type	Article
Publication Date	May 1, 2006
Deposit Date	Oct 12, 2012
Publicly Available Date	Oct 16, 2012
Journal	Journal of Number Theory
Print ISSN	0022-314X
Publisher	Elsevier
Peer Reviewed	Peer Reviewed
Volume	118
Issue	1
Pages	71-89
DOI	https://doi.org/10.1016/j.jnt.2005.08.002
Keywords	Elliptic curve, Primitive divisor, Zsigmondy's Theorem, Somos sequence, Elliptic divisibility sequence, Prime.

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Copyright Statement
NOTICE: this is the author’s version of a work that was accepted for publication in Journal of number theory. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of number theory, 118/7, 2006, 10.1016/j.jnt.2005.08.002