Problem: For a given point $Q:=(x_j,y_j)$ on 2-D,
is there always an elliptic curve $E:=y^2=x^3+ax^2+b$ with a point $P:=(x_i,y_i)$ on it,
such that $nP=Q$ and $(|x_i|)^k >ab$,
where $k\leq \frac{(n-\sqrt2)}{ 2}?$
Here, $n$ is the scalar multiple of point $P:=(x_i,y_i)$. $P, Q \in E(\mathbb{Z})$ and $a, b \in \mathbb{Z}$.
Question: Provided there is always an elliptic curve $y^2=x^3+ax^2+b$ with a point $P$ such that $nP=Q$ ,
is it true that there exists point $P$ such that $(|x_i|)^k >ab$ where $k\leq \frac{(n-\sqrt2)}{ 2}, n>0?$ How to prove or disprove it?
Attempt: So far I found no related results, this is probably due to lack of knowledge of terminology on elliptic curve, I know about Hall-Lang Conjecture, Baker's theorem (which is not helpful in this case).
Post Script: Please, let me know about result, terminology (for searching), method related to my question.