Primitive divisors of elliptic divisibility sequences

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

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
• Electronic ISSN: 1096-1658
• 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.
• Public URL: https://durham-repository.worktribe.com/output/1473085