Suppose we have an equation of two variables $x,y$, for example $E: x^3+y^3=d$, and we have a bijective map \begin{align*} \varphi: E &\to F\\ (x,y) &\mapsto \left(\frac{3d}{x+y},-\frac{9dy}{x+y}\right) \end{align*} where $F$ is the (nonsingular) elliptic curve $F : y^2=x^3-432d^2$. If we prove that $F$ doesn't have integer solutions (or has finitely many) is that sufficient to prove that $E$ doesn't have solutions (or has finitely many solutions)? In other words, does the bijective map $\varphi$ from equation to elliptic curve preserve the structure between them?i don't mean this example i mean in general specialy Integer map .


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