Standard

On some determinants involving Jacobi symbols. / Krachun, Dmitry; Petrov, Fedor; Sun, Zhi Wei; Vsemirnov, Maxim.

в: Finite Fields and Their Applications, Том 64, 101672, 06.2020.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

Harvard

Krachun, D, Petrov, F, Sun, ZW & Vsemirnov, M 2020, 'On some determinants involving Jacobi symbols', Finite Fields and Their Applications, Том. 64, 101672. https://doi.org/10.1016/j.ffa.2020.101672

APA

Krachun, D., Petrov, F., Sun, Z. W., & Vsemirnov, M. (2020). On some determinants involving Jacobi symbols. Finite Fields and Their Applications, 64, [101672]. https://doi.org/10.1016/j.ffa.2020.101672

Vancouver

Krachun D, Petrov F, Sun ZW, Vsemirnov M. On some determinants involving Jacobi symbols. Finite Fields and Their Applications. 2020 Июнь;64. 101672. https://doi.org/10.1016/j.ffa.2020.101672

Author

Krachun, Dmitry ; Petrov, Fedor ; Sun, Zhi Wei ; Vsemirnov, Maxim. / On some determinants involving Jacobi symbols. в: Finite Fields and Their Applications. 2020 ; Том 64.

BibTeX

@article{4325d8d63501458a8b70218a19856b27,
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 = "Funding Information: The work is supported by the NSFC (Natural Science Foundation of China)-RFBR (Russian Foundation for Basic Research) Cooperation and Exchange Program (grants NSFC 11811530072 and RFBR 18-51-53020-GFEN-a). The third author is also supported by the Natural Science Foundation of China (grant 11971222). Publisher Copyright: {\textcopyright} 2020 Elsevier Inc. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",
year = "2020",
month = jun,
doi = "10.1016/j.ffa.2020.101672",
language = "English",
volume = "64",
journal = "Finite Fields and Their Applications",
issn = "1071-5797",
publisher = "Elsevier",

}

RIS

TY - JOUR

T1 - On some determinants involving Jacobi symbols

AU - Krachun, Dmitry

AU - Petrov, Fedor

AU - Sun, Zhi Wei

AU - Vsemirnov, Maxim

N1 - Funding Information: The work is supported by the NSFC (Natural Science Foundation of China)-RFBR (Russian Foundation for Basic Research) Cooperation and Exchange Program (grants NSFC 11811530072 and RFBR 18-51-53020-GFEN-a). The third author is also supported by the Natural Science Foundation of China (grant 11971222). Publisher Copyright: © 2020 Elsevier Inc. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

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

UR - http://www.scopus.com/inward/record.url?scp=85081992971&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2020.101672

DO - 10.1016/j.ffa.2020.101672

M3 - Article

AN - SCOPUS:85081992971

VL - 64

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

M1 - 101672

ER -

ID: 75247951