  Let $E$ be an elliptic curve over the rationals given by an integral
Weierstrass model and let $P$ be a rational point of infinite order. The
multiple $nP$ has the form $(A_n/B_n^2,C_n/B_n^3)$ where $A_n$, $B_n$, $C_n$
are integers with $A_n C_n$ and $B_n$ coprime, and $B_n$ positive. The sequence
$(B_n)$ is called the elliptic divisibility sequence generated by $P$. This
paper is concerned with a question posed in 2007 by Everest, Reynolds and
Stevens: does the sequence $(B_n)$ contain only finitely many perfect powers?
We answer this question positively under three additional assumptions: $P$ is
non-integral, the discriminant of $E$ is positive, and $P$ belongs to the
connected real component of the identity on $E$. Our method attaches to the
problem a Frey curve that is defined over a totally real field of degree at
most $24$, and then makes use of modularity and level lowering arguments.
  We can deduce the same theorem without assuming that the discriminant of $E$
is positive, or assuming that $P$ belongs to the connected real component of
the identity, provided we assume some standard conjectures from the Langlands
programme.
