How would one solve the following quartic Diophantine equation in two variables:
$Ax^4 + Bx^3 + Cx^2 + Dx + Ey^2 + Ey = 0$
where A, B, C, D, E are known integers and $x$, $y$ are unknown integers to be solved?
Thanks,
How would one solve the following quartic Diophantine equation in two variables:
$Ax^4 + Bx^3 + Cx^2 + Dx + Ey^2 + Ey = 0$
where A, B, C, D, E are known integers and $x$, $y$ are unknown integers to be solved?
Thanks,
Write the equation in the form
$\text{quadratic in y } = \text{ quartic in }x.$
If the quartic has a repeated root, then this equation cuts out a curve of (geometric) genus zero, which admits a rational parameterization (if not over $\mathbb Q$ then over an explicit finite extension), and so finding the integral points should be straightforward.
E.g. $y^2 = x^3$ admits the parameterization $x = t^2,y=t^3$, and hence integral solutions are given by $(t^2,t^3)$ with $t$ an integer.
If the quartic does not have a repeated root, then, as noted in the comments, the equation cuts out a curve of (geometric) genus one, and a theorem of Siegel states that it has only finitely many integral solutions.
To learn more about this, you should look at a textbook on elliptic curves, of which there are many available. Silverman's graduate text is the most standard reference, although the book of Silverman and Tate is perhaps a better entry-point if Silverman's text seems at too high a level.