I came across one elliptic equation of the form $y^2 = x^3 + p^2$ being $p$ prime, and taking $p \neq 3$, I want to have more understanding why there is no rational point $x$, for $y = 3p$ or $y = 3p^2$, such that:
$$y^2 = x^3 + p^2$$
I want to know in terms of equivalence classes approach also if possible.
Let $y^2 = x^3 + p^2$;
take $y=3p$ place into equation; $9p^2 = x^3 + p^2$ then $x^3 = 8p^2$ take the cube-root; $x=2\sqrt[3]{p^2}$, this implies there is no rational solution.
take $y=3p^2$ place into equation; $9p^4 = x^3 - p^2$ then we have $$x^3 - (9p^4 + p^2)$$ Look at the discrimant $\Delta_x = -27 (9p^4-p^2)^2$. It is negative, therefore, one real two complex roots. Complext roots cannot be a solution. The real root is $p^{2/3}\sqrt[3]{9p^2-1}$. This concludes that no rational solution here, too.