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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| Published in | Annals of combinatorics Vol. 14; no. 4; pp. 443 - 456 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Basel
SP Birkhäuser Verlag Basel
01.12.2010
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0218-0006 0219-3094 |
| DOI | 10.1007/s00026-011-0069-6 |
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| 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. |
|---|---|
| AbstractList | 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. |
| Author | Eğecioğlu, Ömer |
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| Cites_doi | 10.1515/crll.1933.169.158 10.2140/pjm.1970.34.559 10.4064/aa115-3-4 10.1017/CBO9780511609589 10.1007/978-1-4757-2103-4 |
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| Keywords | prime 11C20 Jacobi symbol 15A36 11T24 Gauss sum determinant Legendre symbol |
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| References | GradshteynI.S.RyzhikI.M.Table of Integrals, Series, and Products1980OrlandoAcademic Press0521.33001 IrelandK.RosenM.A Classical Introduction to Modern Number Theory1990New YorkSpringer-Verlag0712.11001 HardyG.H.WrightE.M.An Introduction to the Theory of Numbers1980New YorkOxford University Press MoonJ.W.Counting Labelled Trees1970Montreal, Que.Canadian Mathematical Congress0214.23204 BorevichZ.I.ShafarevichI.R.Number Theory1966New York-LondonAcademic Press ChapmanR.Determinants of Legendre symbol matricesActa Arith.2004115323124421005021073.1500610.4064/aa115-3-4 WilliamsK.S.Finite transformation formulae involving the Legendre symbolPacific J. Math.1970345595682696180207.05501 DavenportH.On certain exponential sumsJ. Reine Angew. Math.193316915817610.1515/crll.1933.169.158 StanleyR.P.Enumerative Combinatorics, Volume 21999CambridgeCambridge University Press0928.0500110.1017/CBO9780511609589 R. Chapman (69_CR2) 2004; 115 Z.I. Borevich (69_CR1) 1966 I.S. Gradshteyn (69_CR4) 1980 J.W. Moon (69_CR7) 1970 K. Ireland (69_CR6) 1990 G.H. Hardy (69_CR5) 1980 K.S. Williams (69_CR9) 1970; 34 H. Davenport (69_CR3) 1933; 169 R.P. Stanley (69_CR8) 1999 |
| References_xml | – reference: DavenportH.On certain exponential sumsJ. Reine Angew. Math.193316915817610.1515/crll.1933.169.158 – reference: IrelandK.RosenM.A Classical Introduction to Modern Number Theory1990New YorkSpringer-Verlag0712.11001 – reference: BorevichZ.I.ShafarevichI.R.Number Theory1966New York-LondonAcademic Press – reference: StanleyR.P.Enumerative Combinatorics, Volume 21999CambridgeCambridge University Press0928.0500110.1017/CBO9780511609589 – reference: HardyG.H.WrightE.M.An Introduction to the Theory of Numbers1980New YorkOxford University Press – reference: WilliamsK.S.Finite transformation formulae involving the Legendre symbolPacific J. Math.1970345595682696180207.05501 – reference: ChapmanR.Determinants of Legendre symbol matricesActa Arith.2004115323124421005021073.1500610.4064/aa115-3-4 – reference: GradshteynI.S.RyzhikI.M.Table of Integrals, Series, and Products1980OrlandoAcademic Press0521.33001 – reference: MoonJ.W.Counting Labelled Trees1970Montreal, Que.Canadian Mathematical Congress0214.23204 – volume-title: Number Theory year: 1966 ident: 69_CR1 – volume: 169 start-page: 158 year: 1933 ident: 69_CR3 publication-title: J. Reine Angew. Math. doi: 10.1515/crll.1933.169.158 – volume: 34 start-page: 559 year: 1970 ident: 69_CR9 publication-title: Pacific J. Math. doi: 10.2140/pjm.1970.34.559 – volume: 115 start-page: 231 issue: 3 year: 2004 ident: 69_CR2 publication-title: Acta Arith. doi: 10.4064/aa115-3-4 – volume-title: Counting Labelled Trees year: 1970 ident: 69_CR7 – volume-title: An Introduction to the Theory of Numbers year: 1980 ident: 69_CR5 – volume-title: Enumerative Combinatorics, Volume 2 year: 1999 ident: 69_CR8 doi: 10.1017/CBO9780511609589 – volume-title: Table of Integrals, Series, and Products year: 1980 ident: 69_CR4 – volume-title: A Classical Introduction to Modern Number Theory year: 1990 ident: 69_CR6 doi: 10.1007/978-1-4757-2103-4 |
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| Snippet | 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... |
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| StartPage | 443 |
| SubjectTerms | Combinatorics Mathematics Mathematics and Statistics |
| Title | A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators |
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