On generalized Fermat equations of signature $(p,p,3)$
Volume 123 / 2011
Colloquium Mathematicum 123 (2011), 49-52
MSC: Primary 11D41.
DOI: 10.4064/cm123-1-4
Abstract
This paper focuses on the Diophantine equation $x^n+p^{\alpha}y^n=Mz^3$, with fixed $\alpha$, $p$, and $M$. We prove that, under certain conditions on $M$, this equation has no non-trivial integer solutions if $n\geq \mathcal F(M,p^\alpha)$, where $\mathcal F(M,p^{\alpha})$ is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399–1416].
