On the factorization of \(X^ n - BX - A\)
type: Item
instance of: scholarly article
title: On the factorization of \(X^ n - BX - A\)
zbMATH Open document ID: 0758.12002
published in: L'Enseignement Mathématique. 2e Série
publication date: 1992-09-26T00:00:00Z
review text: Generalizing a theorem of \textit{S. Rabinowitz} [Math. Mag. 61, 191-193 (1988; Zbl 0649.10010)] about polynomials \(x^ 5-x+A\), the author proves that if \(n\geq 5\) and either \(A\) or \(B\) is given then there are only finitely many polynomials \(x^ n-Bx-A\) with integer coefficients, divisible by a quadratic polynomial \(x^ 2-bx-a\) with integer coefficients. Moreover (via Thue theorem) these may be explicitly determined. The proof involves studying related recurrence sequences. As an example he notes that \(x^ 5+x+A\) has a quadratic factor for \(A=\pm 1\) or \(\pm 6\), and this corresponds to the fact that 1 and 144 are the only non-zero squares in the Fibonacci sequence.
reviewed by: Andrew Granville
Mathematics Subject Classification ID: 12D05
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 55788
zbMATH Keywords: polynomials with integer coefficients
zbMATH Keywords: divisibility by a quadratic polynomial with integer coefficients
zbMATH Keywords: Thue's theorem
zbMATH Keywords: recurrence sequences
author: Paulo Ribenboim
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 55788