On the equation \(P(x)=n!\) and a question of Erdős
type: Item
instance of: scholarly article
title: On the equation \(P(x)=n!\) and a question of Erdős
zbMATH Open document ID: 0762.11010
DOI: 10.1016/0022-314X(92)90020-P
author: Daniel Berend
author: Charles F. Osgood
published in: Journal of Number Theory
publication date: 1993-01-16T00:00:00Z
review text: 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.
reviewed by: Yaochen Zhu
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 93288
zbMATH Keywords: \(G\)-functions
zbMATH Keywords: polynomial of higher degree
zbMATH Keywords: integer solution
zbMATH Keywords: density zero
zbMATH Keywords: diophantine equations
MaRDI profile type: Publication
cites work: Thue-Siegel-Roth theorem for values of algebraic functions
cites work: Unsolved problems in number theory
cites work: Product type bounds on the approximation of values of E- and G-functions
description: scientific article; zbMATH DE number 93288