On some determinants involving Jacobi symbols

Dmitry Krachun; Fedor Petrov; Zhi Wei Sun; Maxim Vsemirnov

doi:10.1016/j.ffa.2020.101672

On some determinants involving Jacobi symbols

Dmitry Krachun, Fedor Petrov, Zhi Wei Sun, Maxim Vsemirnov

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9 Scopus citations

Abstract

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that (6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0 and (4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0 as conjectured by Sun, where [Formula presented] and [Formula presented] with [Formula presented] the Jacobi symbol. We also prove that (10,9)_p=0 for any prime p≡5(mod12), and [5,5]_p=0 for any prime p≡13,17(mod20), which were also conjectured by Sun. Our proofs involve character sums over finite fields.

Original language	English (US)
Article number	101672
Journal	Finite Fields and their Applications
Volume	64
DOIs	https://doi.org/10.1016/j.ffa.2020.101672
State	Published - Jun 2020
Externally published	Yes

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Algebra and Number Theory
General Engineering
Applied Mathematics

Keywords

Character sums over finite fields
Determinants
Jacobi symbols

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10.1016/j.ffa.2020.101672

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@article{4276985ea0f449bf98117db66f417bb9,

title = "On some determinants involving Jacobi symbols",

abstract = "In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that (6,1)n=[6,1]n=(3,2)n=[3,2]n=0 and (4,2)n=(8,8)n=(3,3)n=(21,112)n=0 as conjectured by Sun, where [Formula presented] and [Formula presented] with [Formula presented] the Jacobi symbol. We also prove that (10,9)p=0 for any prime p≡5(mod12), and [5,5]p=0 for any prime p≡13,17(mod20), which were also conjectured by Sun. Our proofs involve character sums over finite fields.",

keywords = "Character sums over finite fields, Determinants, Jacobi symbols",

author = "Dmitry Krachun and Fedor Petrov and Sun, {Zhi Wei} and Maxim Vsemirnov",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2020",

month = jun,

doi = "10.1016/j.ffa.2020.101672",

language = "English (US)",

volume = "64",

journal = "Finite Fields and their Applications",

issn = "1071-5797",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - On some determinants involving Jacobi symbols

AU - Krachun, Dmitry

AU - Petrov, Fedor

AU - Sun, Zhi Wei

AU - Vsemirnov, Maxim

PY - 2020/6

Y1 - 2020/6

N2 - In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that (6,1)n=[6,1]n=(3,2)n=[3,2]n=0 and (4,2)n=(8,8)n=(3,3)n=(21,112)n=0 as conjectured by Sun, where [Formula presented] and [Formula presented] with [Formula presented] the Jacobi symbol. We also prove that (10,9)p=0 for any prime p≡5(mod12), and [5,5]p=0 for any prime p≡13,17(mod20), which were also conjectured by Sun. Our proofs involve character sums over finite fields.

AB - In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that (6,1)n=[6,1]n=(3,2)n=[3,2]n=0 and (4,2)n=(8,8)n=(3,3)n=(21,112)n=0 as conjectured by Sun, where [Formula presented] and [Formula presented] with [Formula presented] the Jacobi symbol. We also prove that (10,9)p=0 for any prime p≡5(mod12), and [5,5]p=0 for any prime p≡13,17(mod20), which were also conjectured by Sun. Our proofs involve character sums over finite fields.

KW - Character sums over finite fields

KW - Determinants

KW - Jacobi symbols

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DO - 10.1016/j.ffa.2020.101672

M3 - Article

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SN - 1071-5797

VL - 64

JO - Finite Fields and their Applications

JF - Finite Fields and their Applications

M1 - 101672

ER -

On some determinants involving Jacobi symbols

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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