Arithmetical properties of wendt's determinant

✍ Scribed by Charles Helou; Guy Terjanian

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 161 KB
• Volume: 115
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.09.003

✦ Synopsis

Wendt's determinant of order n is the circulant determinant W n whose (i, j )-th entry is the binomial coefficient n |i-j | , 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 W p k ≡ 1 (mod p k ) and W np k ≡ W n (mod p). If q is another prime, distinct from p, and h any positive integer, then W p h q k ≡ W p h W q k (mod pq). Furthermore, if p is odd, then W p ≡ 1 + p 2p-1 p-1 -1 (mod p 5 ). In particular, if p 5, then W p ≡ 1 (mod p 4 ). Also, if m and n are relatively prime positive integers, then W m W n divides W mn .