I'm a question about Nagell-Nutz theorem. For example, the point $P'=( \frac{31073}{2704},-\frac{5491823}{140608})$ on the curve $$y^2=x^3+8$$ does not meet the theorem Nagell-Nutz. So I can say what the point $P=(2,4)$ has infinite order by the Nagell-Lutz theorem? Since $P'=4P$.
I am not fluent in English. Sorry for any error.
Thank you very much!
Yes, that conclusion is correct. By Nagell-Lutz the points of finite order must have integer coordinates, thus $P'$ has infinite order and by the relation $4P = P'$ we conclude that $P$ has infinite order too.