On the Diophantine equation \(x (x + 1) \dots (x + n) + 1 = y^2\)
type: Item
instance of: scholarly article
title: On the Diophantine equation \(x (x + 1) \dots (x + n) + 1 = y^2\)
zbMATH Open document ID: 0996.11022
author: Nobuhisa Abe
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-11-04T00:00:00Z
review text: The author determines all the integer solutions of the title equation for \(1\leq n\leq 15\), \(n\) odd. His method eventually relies on Runge's method (see e.g. the paper of \textit{A. Grytczuk} and \textit{A. Schinzel} [Colloq. Math. Soc. János Bolyai 60, 329-356 (1992; Zbl 0849.11033)] and the references given there, or the paper of \textit{P. G. Walsh} [Acta Arith. 62, 157-172 (1992; Zbl 0769.11017)] for a quantitative variant of the method). For \(n\) odd, the title equation is a special case of \[ f(x)= g(y) \quad\text{in }x,y\in \mathbb{Z}, \tag{1} \] where \(f,g\in \mathbb{Z}[x]\) are monic polynomials, and \(\deg(g)\mid \deg(f)\). The method of solution of (1) was completely described by \textit{L. Szalay} [Superelliptic equations of the form \(y^p= x^{kp}+ a_{kp-1} x^{kp-1}+\cdots+ a_0\), Bull. Greek Math. Soc. (to appear)] and the case when \(\deg(g)= 2\), earlier by \textit{L. Szalay} [Acta Acad. Paedagog. Agriensis, Sect. Mat. 27, 19-24 (2000; Zbl 0973.11039)]. The actual way of treatment in the latter paper is very similar to the one followed by the author. None of the above or other relevant references are cited in the article; the author only refers to the well-known paper of \textit{P. Erdős} and \textit{J. L. Selfridge} [Ill. J. Math. 19, 292-301 (1975; Zbl 0295.10017)].
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 1471662
zbMATH Keywords: consecutive integers
zbMATH Keywords: Runge's method
reviewed by: Lajos Hajdu
MaRDI profile type: Publication
full work available at URL: pjaa.76.16
OpenAlex ID: W2041772319
cites work: The product of consecutive integers is never a power
DOI: 10.3792/PJAA.76.16
description: scientific article; zbMATH DE number 1471662