Vol. 12, No. 3, 2019

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An elliptic curve analogue to the Fermat numbers

Skye Binegar, Randy Dominick, Meagan Kenney, Jeremy Rouse and Alex Walsh

Vol. 12 (2019), No. 3, 427–449
Abstract

The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use rational points of infinite order on elliptic curves 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.

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Keywords
elliptic curves, Fermat numbers, duplication formula
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 11B37, 11G15, 11Y11
Milestones
Received: 12 August 2017
Revised: 17 July 2018
Accepted: 22 July 2018
Published: 14 December 2018

Communicated by Bjorn Poonen
Authors
Skye Binegar
Department of Mathematics
Reed College
Portland, OR
United States
Randy Dominick
Department of Mathematics & Statistics
Texas Tech University
Lubbock, TX
United States
Meagan Kenney
Department of Mathematics
Bard College
Annandale-on-Hudson, NY
United States
Jeremy Rouse
Department of Mathematics and Statistics
Wake Forest University
Winston-Salem, NC
United States
Alex Walsh
Mathematics Department
Brown University
Providence, RI
United States