  Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not
contain the subfield of $\mathbb{Q} (\zeta_{p^2})$ of degree $p$ over
$\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$.
We prove that the existence of a counterexample to the local-global
divisibility by $p^2$ in $\mathcal{E}$, assures the existence of a $k$-rational
point of exact order $p$ in $\mathcal{E}$. Using the Merel Theorem, we then
shrunk the known set of primes for which there could be a counterexample to the
local-global divisibility by $p^2$.
