  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.
