Arithmetical properties of Wendt's determinant
type: Item
instance of: scholarly article
title: Arithmetical properties of Wendt's determinant
zbMATH Open document ID: 1089.11018
author: Charles Helou
author: Guy Terjanian
published in: Journal of Number Theory
publication date: 2006-03-08T00:00:00Z
review text: Wendt's determinant \(W_n\) is the determinant of the \(n\times n\) circulant matrix, whose \((i, j)\) entry is the binomial coefficient \(n\choose | i-j| \). It also can be defined as the resultant of the polynomials \(X^n -1\) and \((X+1)^n-1\) or via cyclotomic fields. Most of the investigations on Wendt's determinant have focused on determining its prime factorization. The present authors derive congruence relations for the numbers \(W_n\), making use of resultants and cyclotomic fields. For example, they show that if \(m\) and \(n\) are relatively prime then \(W_m W_n\) divides \(W_{mn}\), that \(W_p\equiv 1+p({2p-1\choose p-1}-1)\pmod{p^5}\) for \(p\geq 3\) and, in particular, that \(W_p\equiv 1\pmod{p^4}\) for \(p\geq 5\). Wendt's determinant was initially introduced and studied in connection with Fermat's last theorem.
reviewed by: Pentti Haukkanen
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5009835
zbMATH Keywords: Wendt's determinant
zbMATH Keywords: congruence
zbMATH Keywords: circulant matrix
zbMATH Keywords: binomial coefficient
zbMATH Keywords: resultant
zbMATH Keywords: cyclotomic field
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.09.003
OpenAlex ID: W2018848441
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DOI: 10.1016/J.JNT.2004.09.003
description: scientific article; zbMATH DE number 5009835