Let $E$ be an elliptic curve given by an equation $E : y^2 = x^3 + Ax + B$ with $A, B \in Q$. Then the group of rational points E(Q) is a finitely generated abelian group. In other words, there is a finite set of points P1, . . . , Pt ∈ E(Q) so that every point $P ∈ E(Q)$ can be written in the form $P = n_1P_1 + n_2P_2 + · · · + n_tP_t$ for some $n_1, n_2, . . . , n_t ∈ Z$. This is Mordell's Theorem (1922).
Does above theorem guarantees that $nP=Q$ for given $P, Q$, where $P,Q \in E$ ?
If not then what is the condition for $n$ such that $nP=Q$ for $P,Q \in E$?