On local-global divisibility by $p^n$ in elliptic curves

Let $p$ be a prime number and let $k$ be a number field, which does not contain the field $\mathbb{Q}({\zeta }_p+\overline{{\zeta }_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion points of exact order $p$ on $\text{E}$, then the local-global principle holds for divisibility by $p^n$, with $n$ a natural number. As a consequence of the deep theorem of Merel, for $p$ larger than a constant depending only on the degree of $k$, there are no counterexamples to the local-global divisibility principle. Nice and deep works give explicit small constants for elliptic curves defined over a number field of degree at most 5 over $$\mathbb{Q}$.