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On Wendt's determinant

Author: Charles Helou
Journal: Math. Comp. 66 (1997), 1341-1346
MSC (1991): Primary {11C20; Secondary 11Y40, 11D41, 12E10}
MathSciNet review: 1423075
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Abstract: Wendt's determinant of order $m$ is the circulant determinant $W_{m}$ whose $(i,j)$-th entry is the binomial coefficient $\binom m{|i-j|}$, for $1\leq i,j\leq m$. We give a formula for $W_{m}$, when $m$ is even not divisible by 6, in terms of the discriminant of a polynomial $T_{m+1}$, with rational coefficients, associated to $(X+1)^{m+1}-X^{m+1}-1$. In particular, when $m=p-1$ where $p$ is a prime $\equiv -1\ (mod\ 6)$, this yields a factorization of $W_{p-1}$ involving a Fermat quotient, a power of $p$ and the 6-th power of an integer.

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Charles Helou
Affiliation: Penn State University, Delaware County, 25 Yearsley Mill Road, Media, Pennsylvania 19063

Received by editor(s): May 6, 1996
Article copyright: © Copyright 1997 American Mathematical Society