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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?


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Already the case $A=0$ is an enormous subject (elliptic curves). – André Nicolas Mar 6 '12 at 13:21
Good to know. Thanks. – SPSmith Mar 6 '12 at 15:28
In fact, when $A\neq 0$, this is still an elliptic curve. See L. Washington, Elliptic Curves: Number Theory and Cryptography, page 37. – Álvaro Lozano-Robledo Mar 6 '12 at 20:46
up vote 6 down vote accepted

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.

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