Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant Wn 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 Wpk≡1(modpk) and Wnpk≡Wn(modp). If q is another prime, distinct from p, and h any positive integer, then Wphqk≡WphWqk(modpq). Furthermore, if p is odd, then Wp≡1+p2p-1p-1-1(modp5). In particular, if p⩾5, then Wp≡1(modp4). Also, if m and n are relatively prime positive integers, then WmWn divides Wmn.