On the equation \(f(x+1)\dots f(x+k)= f(y+1)\dots f(y+mk)\)
type: Item
instance of: scholarly article
title: On the equation \(f(x+1)\dots f(x+k)= f(y+1)\dots f(y+mk)\)
zbMATH Open document ID: 0795.11015
DOI: 10.1016/0019-3577(93)90001-F
author: R. Balasubramanian
author: T. N. Shorey
published in: Indagationes Mathematicae. New Series
publication date: 1994-03-01T00:00:00Z
review text: The authors study the integral solutions of the equation \[ f(x+1)\cdot\dots\cdot f(x+k)= f(y+1)\cdot \dots\cdot f(y+m k) \tag{*} \] when \(f\) is a power of a monic irreducible polynomial with rational coefficients and \(k\geq 2\), \(m\geq 2\) are integral parameters. A special case is \(f(x)=x\), previously studied by \textit{N. Saradha} and the second named author [Indag. Math., New Ser. 3, 79-90 (1992; Zbl 0757.11011)]. In that paper it was proved that, for integers \(x\geq 0\), \(y\geq 0\), \(k\geq 2\) and \(m\geq 2\), the equation \((x+1)\dots (x+k)= (y+1)\dots (y+mk)\) implies that \(\max(x,y,k)\) is bounded by an effective constant depending only on \(m\). In the present paper results of an analogous type are proved for equation (*), provided that certain reasonable assumptions are fulfilled. Remark. In the reviewer's opinion, it is worthwhile in papers like the one under review to -- at least -- discuss the possibility of making the stated results explicit.
Mathematics Subject Classification ID: 11D57
zbMATH DE Number: 510025
zbMATH Keywords: integral solutions
zbMATH Keywords: polynomial with rational coefficients
reviewed by: Nikos Tzanakis
MaRDI profile type: Publication
cites work: On the equation \((x+1)\dots(x+k)=(y+1)\dots(y+mk)\)
cites work: Die Primteilerdichte von ganzzahligen Polynomen. I.
full work available at URL: 0019-3577(93)90001-f
OpenAlex ID: W2025305059
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description: scientific article; zbMATH DE number 510025