Polynomial values of sums of products of consecutive integers
type: Item
instance of: scholarly article
title: Polynomial values of sums of products of consecutive integers
zbMATH Open document ID: 1442.11055
published in: Monatshefte für Mathematik
publication date: 2018-09-14T00:00:00Z
full work available at URL: BBHL-SumsOfProducts.pdf
review text: Following \textit{L. Hajdu} et al. [Acta Arith. 172, No. 4, 333--349 (2016; Zbl 1400.11087)], for \(k=0,1,2,\ldots \) let \[f_k(x)=\sum_{i=0}^k\prod_{j=0}^i (x+j).\] Let \(g(x)\) be an arbitrary polynomial with rational coefficients. Consider the equation \[f_k(x)=g(y)\;\; \text{in} \;\;\ x,y\in\mathbb Z.\] In case the degree of \(g\) is 0 or 2, the authors show that an effectively computable upper bound can be derived for the absolute values of the solutions. If the degree of \(g\) is \(\geq 3\), then it is shown, that the equation has only finitely many solutions, unless \(g(x)=f_k(h(x))\) with a polynomial \(h\) with rational coefficients and positive degree. In the proof of the second statement an ineffective theorem of \textit{Y. F. Bilu} and \textit{R. F. Tichy} [Acta Arith. 95, No. 3, 261--288 (2000; Zbl 0958.11049)] is applied.
reviewed by: István Gaál
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 6936067
zbMATH Keywords: sums of products
zbMATH Keywords: blocks of consecutive integers
zbMATH Keywords: polynomial values
author: András Bazsó
author: Attila Bérczes
author: Lajos Hajdu
author: Florian Luca
MaRDI profile type: Publication
OpenAlex ID: W2766975353
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DOI: 10.1007/S00605-017-1130-2
description: scientific article; zbMATH DE number 6936067