Elliptic Fermat numbers and elliptic divisibility sequence

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.