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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory