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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, 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?

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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 $p$ satisfies 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?

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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.group.(The curve is quasi-minimal,say,there are no primes $p$ satisfies $p^4\vert a$ 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?

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