A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators

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...

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Bibliographic Details
Published inAnnals of combinatorics Vol. 14; no. 4; pp. 443 - 456
Main Author Eğecioğlu, Ömer
Format Journal Article
LanguageEnglish
Published Basel SP Birkhäuser Verlag Basel 01.12.2010
Subjects
Combinatorics
Mathematics
Mathematics and Statistics
prime
11C20
Jacobi symbol
15A36
11T24
Gauss sum
determinant
Legendre symbol
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ISSN0218-0006
0219-3094
DOI10.1007/s00026-011-0069-6

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Summary: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.
ISSN:0218-0006
0219-3094
DOI:10.1007/s00026-011-0069-6