An elliptic curve analogue to the Fermat numbers

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 01 Jan 2019Embargo end date: 01 Aug 2017 English Publisher:Mathematical Sciences PublishersJournal:Involve, a Journal of Mathematics, volume 12, pages 427-449 (issn: 1944-4176, eissn: 1944-4184,

Copyright policy )Funded by:NSF | REU Site: Wake/Davidson E...

Authors: Binegar, Skye; Dominick, Randy; Kenney, Meagan; Rouse, Jeremy; Walsh, Alex;

doi: 10.2140/involve.2019.12.427 , 10.48550/arxiv.1708.03804

arXiv: 1708.03804

An elliptic curve analogue to the Fermat numbers

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Abstract

The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are analogous to the Fermat numbers. We demonstrate that these sequences have many of the same properties as the Fermat numbers, and we discuss results about the prime factors of sequences generated by specific curves and points.

24 pages

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Keywords

Primality, Fermat numbers, Primary 11G05, Secondary 11B37, 11G15, 11Y11, Mathematics - Number Theory, duplication formula, 11Y11, Complex multiplication and moduli of abelian varieties, 11B37, Elliptic curves over global fields, elliptic curves, FOS: Mathematics, Recurrences, 11G15, Number Theory (math.NT), 11G05

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