Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W_n 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 W_pk ≡ 1 (mod p^k) and W_npk ≡ W_n (mod p). If q is another prime, distinct from p, and h any positive integer, then W_phqk ≡ W_phqk (mod pq). Furthermore, if p is odd, then W_p ≡ 1 + p ((_p-1^2p-1) - 1) (mod p⁵). In particular, if p ≥ 5, then W_p ≡ 1 (mod p⁴). Also, if m and n are relatively prime positive integers, then W_m W_n divides W_mn.