On some determinants involving Jacobi symbols

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

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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 languageEnglish (US)
Article number101672
JournalFinite Fields and their Applications
Volume64
DOIs
StatePublished - Jun 2020
Externally publishedYes

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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