On the equation \(P(x)=n!\) and a question of Erdős
Let \(P\) be a polynomial of degree \(\geq 2\) over \(\mathbb{Z}\) and \(s\) be a fixed non-zero integer. The authors prove that the set \(\{n\mid\) \(n\in\mathbb{N}\), \(P(x)=s\cdot n!\) has a solution \(x\in\mathbb{Z}\}\) is of density zero. This answers in particular a question of Erdős, which originated from a study of diophantine equations \(x^ p\pm y^ p=n!\) [\textit{P. Erdős} and \textit{R. Obláth}, Acta Litt. Sci. Szeged. 8, 241-255 (1937; Zbl 0017.00404)]. The proof of this theorem bases on results concerning \(G\)-functions, and is very enlightening.
scientific article; zbMATH DE number 93288