I am interested in how to tackle this Diophantine equation:
The solutions I have found so far are $(1,1)$ and $(7,37)$. Are there any more?
I have looked up various material on cubic Diophantines but most of what I’ve found is on equations where the coefficients of $x^3$ and $y^2$ are the same. In this particular problem, if both coefficients were equal to $1$, it would just be a nice Mordell’s equation. But the coefficient of the cubic variable is not $1$ – which is why it’s so frustrating. Still, would I be right in saying that if the solutions were to lie on an elliptic curve, there would only be finitely many of them? What if they don’t lie on an elliptic curve? Will the number of solutions still be finite?