Rational points which are hard to find are those of large height, and in particular large denominator. This method will only find rational points with denominator u when you scale the equations by u, which requires knowing u in advance or looping over all possible u. Try this curve: https://www.lmfdb.org/EllipticCurve/Q/294504803/d/1


For what it's worth, it should not be possible to make $P$ and $nP$ integral on a quasi-minimal equation if $n$ is very large. More precisely, the following theorem holds. It quantifies what Chris Wuthrich said in the comments. Theorem: If Lang's height conjecture is true (or if the $ABC$-conjecture is true), then there is an absolute constant $C$ such if $E/...

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