Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and
$$G=\{P,2P,3P,4P,\cdots\}.$$
Is there an integral point $Q\in G?$



Is there an integral point in the group generated by an rational point?
I presume that (1) $E$ is meant to be in minimal form with integer coefficients,
else you can change coordinates to put $P$ and $nP$ anywhere, and that
(2) $P$ is not itself integral (i.e. $d>1$), else $Q=P$ works.


Then the answer is No, because $d^2$ and $d^3$ divide the denominators
of $x(nP)$ and $y(nP)$ for all $n$. 


This is a known property of "elliptic divisibility sequences".
One way to think about it is that $p|d$ means that $P$ is $p$-adically
near the identity point $\infty$ on $E$ (and $p^e|d$ for $e>1$ makes $P$ even nearer),
and then any multiple of $P$ is at least as close.


