An Elliptic Curve Analogue to the Fermat Numbers

TL;DR

This paper introduces a new class of sequences derived from elliptic curves that mimic Fermat numbers, sharing key properties like coprimality and recurrence, and explores their prime factorization characteristics.

Contribution

It extends Fermat number properties to sequences generated by elliptic curves using rational points of infinite order, a novel approach in number theory.

Findings

Sequences share properties with Fermat numbers such as coprimality and recurrence.

Results on prime factors of sequences from specific elliptic curves.

Demonstrates the potential of elliptic curves to generate Fermat-like sequences.

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.