Arithmetical properties of Wendt's determinant (Q817249)

Wendt's determinant \(W_n\) is the determinant of the \(n\times n\) circulant matrix, whose \((i, j)\) entry is the binomial coefficient \(n\choose | i-j| \). It also can be defined as the resultant of the polynomials \(X^n -1\) and \((X+1)^n-1\) or via cyclotomic fields. Most of the investigations on Wendt's determinant have focused on determining its prime factorization. The present authors derive congruence relations for the numbers \(W_n\), making use of resultants and cyclotomic fields. For example, they show that if \(m\) and \(n\) are relatively prime then \(W_m W_n\) divides \(W_{mn}\), that \(W_p\equiv 1+p({2p-1\choose p-1}-1)\pmod{p^5}\) for \(p\geq 3\) and, in particular, that \(W_p\equiv 1\pmod{p^4}\) for \(p\geq 5\). Wendt's determinant was initially introduced and studied in connection with Fermat's last theorem.