Consider an elliptic curve $E:y^2=x^3+ax+b,a,b\in\mathbb{Z}$ on $\mathbb{Q}$ with rank $r$ and trivial torsion group.(**The curve is quasi-minimal**,say,there are no primes $p$ such that $p^4\vert a$ and $p^6\vert b$)

Question:Suppose this curve has **many integer points**(say, more than $2^{r+1}$ integer points with $y>0$),is it possible to find $r$ integer points to generate the whole Mordell-Weil Group $E(\mathbb{Q})$?If not,is it possible to give an counterexample?

`$#E(\mathbb{Z})\ll C^{rank E(\mathbb{Q}}$`

(for quasi-minimal Weierstrass equations) is not so easy to prove. (At least, Marc Hindry and I didn't find it so easy!) $\endgroup$3more comments