On some determinants involving Jacobi symbols

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

In: Finite Fields and Their Applications, Vol. 64, 101672, 06.2020.

Research output: Contribution to journal › Article › peer-review

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