Diophantine equations with products of consecutive values of a quadratic polynomial
type: Item
instance of: scholarly article
title: Diophantine equations with products of consecutive values of a quadratic polynomial
zbMATH Open document ID: 1242.11023
author: Bo He
author: Shichun Yang
author: Alain Togbé
published in: Journal of Number Theory
publication date: 2011-07-22T00:00:00Z
review text: Let \(a,b,c,d\) be fixed nonnegative integers with \(a,d \geq 1\) and \(\gcd (a,b,c)=1\). Suppose that \(f(X)=aX^2+bX+c\) is an irreducible polynomial over \(\mathbb Q\). The authors prove that the solutions of the Diophantine equation \[ \prod_{k=1} ^n f(k)=dy^l, \quad l,n,y \in {\mathbb N}, \;l\geq 2, \] verify \[ n\leq \max \{ 10^8, b,c,\frac{d}{2a}, n_0\}, \] where \(n_0\) is explicitly determined in terms of \(a\), \(b\), \(c\). One of the consequences of this result is that the only positive integer solution to the equation \(\prod_{k=1} ^n (2k^2+7)=5y^l\) is \((n,y,l)=(2,3,3)\). The proof uses the idea of \textit{J. Cilleruelo} [J. Number Theory 128, No. 8, 2488--2491 (2008; Zbl 1213.11057)] and various inequalities concerning the distribution of prime numbers.
reviewed by: Mihai Cipu
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 5931158
zbMATH Keywords: quadratic polynomials
zbMATH Keywords: squares
zbMATH Keywords: powers
zbMATH Keywords: Chebyshev inequalities for primes
zbMATH Keywords: arithmetic progressions of prime numbers
MaRDI profile type: Publication
full work available at URL: j.jnt.2011.03.006
OpenAlex ID: W2013765233
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DOI: 10.1016/J.JNT.2011.03.006
description: scientific article; zbMATH DE number 5931158