Elliptic Fermat numbers and elliptic divisibility sequence

TL;DR

This paper introduces elliptic Fermat numbers derived from elliptic divisibility sequences, explores their properties, and demonstrates that only finitely many are prime in certain cases, extending classical Fermat number properties.

Contribution

It defines elliptic Fermat numbers and generalized versions, proving properties like coprimality and compositeness, and shows finitely many primes occur in magnified sequences.

Findings

Finitely many prime terms in magnified elliptic Fermat sequences

Classical Fermat properties extend to generalized elliptic Fermat numbers

Elliptic Fermat numbers share properties with classical Fermat numbers

Abstract

For a pair $(E,P)$ of an elliptic curve $E/\mathbb{Q}$ and a nontorsion point $P\in E(\mathbb{Q})$, the sequence of \emph{elliptic Fermat numbers} is defined by taking quotients of terms in the corresponding elliptic divisibility sequence $(D_{n})_{n\in\mathbb{N}}$ with index powers of two, i.e. $D_{1}$, $D_{2}/D_{1}$, $D_{4}/D_{2}$, etc. Elliptic Fermat numbers share many properties with the classical Fermat numbers, $F_{k}=2^{2^k}+1$. In the present paper, we show that for magnified elliptic Fermat sequences, only finitely many terms are prime. We also define \emph{generalized elliptic Fermat numbers} by taking quotients of terms in elliptic divisibility sequences that correspond to powers of any integer $m$, and show that many of the classical Fermat properties, including coprimality, order universality and compositeness, still hold.