Arithmetical properties of wendt's determinant

Charles Helou, Guy Terjanian

Research output: Contribution to journalArticle

Abstract

Wendt's determinant of order n is the circulant determinant Wn whose (i,j)-th entry is the binomial coefficient ( i-jn , for 1≤i, j≤n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then Wpk ≡ 1 (mod pk) and Wnpk ≡ Wn (mod p). If q is another prime, distinct from p, and h any positive integer, then Wphqk ≡ Wphqk (mod pq). Furthermore, if p is odd, then Wp ≡ 1 + p ((p-12p-1) - 1) (mod p5). In particular, if p ≥ 5, then Wp ≡ 1 (mod p4). Also, if m and n are relatively prime positive integers, then Wm Wn divides Wmn.

Original languageEnglish (US)
Pages (from-to)45-57
Number of pages13
JournalJournal of Number Theory
Volume115
Issue number1
DOIs
StatePublished - Nov 1 2005

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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