On squares in polynomial products
type: Item
instance of: scholarly article
title: On squares in polynomial products
zbMATH Open document ID: 1273.11124
author: Javier Cilleruelo
author: Adolfo Quirós
author: Igor E. Shparlinski
author: Florian Luca
published in: Monatshefte für Mathematik
publication date: 2010-02-23T00:00:00Z
full work available at URL: http://hdl.handle.net/10261/31044
review text: Let \(f(X)\in \mathbb{Z}[X]\) be an irreducible polynomial of degree \(D\geq 2\), and let \(n\) be a positive integer. Define \[ F(n)=\prod_{m=1}^{n}f(m). \] For squarefree integer \(d\), nonnegative integers \(M\) and \(N\), we let \(S_d(M,N)\) be the number of integer solutions \((n,s)\) to the equation \[ F(n)=ds^2,\qquad \text{for}\quad n=M+1,\cdots, M+N. \] This paper proved that \[ S_d(M,N)\ll N^{11/12}(\log N)^{1/3} \] uniformly for squarefree integers \(d\geq 1\) and arbitrary integers \(M\geq 0\) and \(N\geq 2\). The above result yields a lower bound on the number of distinct quadratic fields among \(\mathbb{Q}(\sqrt{F(n)})\) for \(n=M+1,\cdots, M+N\).
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 5674289
zbMATH Keywords: quadratic fields
zbMATH Keywords: square sieve
zbMATH Keywords: character sums
MaRDI profile type: Publication
OpenAlex ID: W2029966005
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DOI: 10.1007/S00605-008-0066-Y
description: scientific article; zbMATH DE number 5674289