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A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators
Ömer Egecioglu
Department of Computer Science, University of California, Santa Barbara, CA 93106, USA
omer@cs.ucsb.edu
Annals of Combinatorics 14 (4) pp.443-456 December, 2010
AMS Subject Classification: 11C20, 15A36, 11T24
Abstract:
We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p-1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of -1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.
Keywords: determinant, prime, Legendre symbol, Jacobi symbol, Gauss sum

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