On Wendt's determinant

Wendt’s determinant of order m is the circulant determinant Wm whose (i, j)-th entry is the binomial coefficient ( m |i−j| ) , 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+1, 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. Introduction E. Wendt ([12]) introduced the m×m circulant determinant Wm with first row the binomial coefficients ( m 0 ) , ( m 1 ) , . . . , ( m m−1 ) , i.e. Wm = ∣∣∣∣∣∣∣ 1 ( m 1 ) ( m 2 ) . . . ( m m−1 ) ( m m−1 ) 1 ( m 1 ) . . . ( m m−2 ) .. .. .. .. ( m 1 ) ( m 2 ) ( m 3 ) . . . 1 ∣∣∣∣∣∣∣ , which is the resultant of the polynomials X − 1 and (X + 1) − 1, in connection with Fermat’s last theorem ([10]). E. Lehmer ([9]) proved that Wm = 0 if and only if m ≡ 0 (mod 6), and that if p is an odd prime number, then Wp−1 is divisible by pp−2qp(2), where qp(2) = 2 p−1−1 p is a Fermat quotient. L. Carlitz ([2]) determined Wp−1 modulo pp−1, which he then used to find high powers of p dividing Wp−1 in an application in the same connection ([3]). Factorizations of the integers Wm for m ≤ 50 were given in ([7]). The size of Wm was investigated in ([1]). Granville and Fee ([5]) determined the prime factors of Wm for all even m ≤ 200 and consequently improved on a classical result about Fermat’s equation. This was further improved in ([6]), where similar computations were carried up to m ≤ 500. In this article, we show that for all positive even integers m not divisible by 6, Wm = −9hm(2m − 1)(m+ 1)m−4|hm|D6 m, where Dm is the discriminant of a polynomial with rational coefficients whose roots are given by a rational function of those of (X + 1) −Xm+1 − 1, and hm = 2 or −1 according as m ≡ 2 or 4 (mod 6) respectively. In particular, if p is a prime ≡ −1 (mod 6) then Dp−1 is a rational integer and we have the factorization Wp−1 = − 9 qp(2) 3pp−2D6 p−1. Received by the editor May 6, 1996. 1991 Mathematics Subject Classification. Primary 11C20; Secondary 11Y40, 11D41, 12E10. c ©1997 American Mathematical Society