Wendt's determinant of order m is the circulant determinant Wm whose (i, j)-th entry is the binomial coefficient (|i-j|m), for 1 ≤ i, j ≤ m. We give a formula for Wm, when m, is even not divisible by 6, in terms of the discriminant of a polynomial Tm+i, with rational coefficients, associated to (X + 1)m+1 - Xm+1 - 1. In particular, when m = p - 1 where p is a prime ≡ - 1 (mod 6), this yields a factorization of Wp-1 involving a Fermat quotient, a power of p and the 6-th power of an integer.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics