On a sequence of polynomials
type: Item
instance of: scholarly article
title: On a sequence of polynomials
zbMATH Open document ID: 0876.11009
DOI: 10.1016/S0022-4049(97)00006-6
author: Frits Beukers
published in: Journal of Pure and Applied Algebra
publication date: 1997-11-26T00:00:00Z
review text: For any integer \(k\geq 2\) let \(p_k\) be a polynomial defined by \(p_k(X)= (X+1)^k -X^k-1\). It is easy to see that \(p_k\) is divisible by \(X\), \(X(X+1)\), \(X(X^2 +X+1)\) and \(X(X+1)\) \((X^2+ X+1)^2\) if \(k\) is even, \(k\equiv 3\pmod 6\), \(k\equiv 5\pmod 6\) and \(k\equiv 1 \pmod 6\), respectively. In connection with studies of solutions to the Korteweg-de Vries equation, J. Sanders asked if, for any \(l>k>1\), \(\text{gcd} (p_k,p_l)\) contains only the trivial divisors mentioned above. In this paper the author proposes a positive reply to this question.
reviewed by: Péter Kiss
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 1025691
zbMATH Keywords: polynomials
zbMATH Keywords: factorisation
zbMATH Keywords: greatest common divisor
zbMATH Keywords: Korteweg-de Vries equation
MaRDI profile type: Publication
cites work: The equation x+y=1 in finitely generated groups
cites work: Lower bounds of heights of points on hypersurfaces
cites work: Linear forms in two logarithms and interpolation determinants
cites work: Algebraic Numbers Close to Both 0 and 1
full work available at URL: s0022-4049(97)00006-6
OpenAlex ID: W2010745556
description: scientific article; zbMATH DE number 1025691