In searching for integral points on elliptic curves, I am encountering Diophantine equations of the following forms: $3m^3 - n^3 = {2^a}{3^b}$, $4m^3 - n^3 = {2^a}{3^b}{5^c}$, $5m^3 - n^3 = {2^a}{3^b}{5^c}$, etc. Are there any papers that bound the number of integer solutions of such Diophantine equations? Here $a$, $b$, and $c$ are positive integers. Thanks to an amazing response by Manuel Norman responding to an earlier question of mine I can conclude that number of solutions is finite. Any references that address such Diophantine equations would be much appreciated.
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Let $f\in\mathbb{Z}[x,y]$ be homogenous form of degree $\geq 3$. Then, the Diophantine equation of the form $F(x,y)=ap_{1}^{\alpha_{1}}\cdots p_{k}^{\alpha_{k}}$, where $a$ is a given non-zero integer and $p_{1}, \ldots, p_{k}$ are fixed primes, has only finitely many solutions in co-prime integers $x, y$. This is a famous result of Mahler which extends classical work of Thue. I encourage you to consult the web-page https://encyclopediaofmath.org/wiki/Thue-Mahler_equation to learn more on the topic and consult reference given therain.
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3$\begingroup$ There's a recent preprint of Adela Gherga and Samir Siksek that gives an efficient algorithm to find all solutions to a Thue-Mahler equation. $\endgroup$ Commented May 7, 2023 at 10:59