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PreprintCC BY 4.0arXiv

Two Legendre-symbol determinants of Sun and Bernoulli numbers

  • Hang Liu
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Abstract

Let p>5 be an odd prime, put n=(p−1)/2, and let (·/p) be the Legendre symbol. Define Dp± to be the following determinants Dp±&:=det[(i± j)((i± j)/p) ]1≤ i,j≤ n. In this article, we give explicit formulas for these determinants modulo p in terms of Bernoulli numbers. As a consequence of Reinhart's recent counterexample to the Ankeny--Artin--Chowla conjecture, we find a prime p such that Dp± ≡ 0 pmod p. This gives a negative answer to a question of the Sun.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026legendresymbol,
  title = {{Two Legendre-symbol determinants of Sun and Bernoulli numbers}},
  author = {Hang Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23684},
  eprint = {2609.23684},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23684},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Two Legendre-symbol determinants of Sun and Bernoulli numbers
AU  - Hang Liu
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23684
UR  - https://arxiv.org/abs/2609.23684
AB  - Let p>5 be an odd prime, put n=(p−1)/2, and let (·/p) be the Legendre symbol. Define D_p^± to be the following determinants D_p^±&:=det[(i± j)((i± j)/p) ]_1≤ i,j≤ n. In this article, we give explicit formulas for these determinants modulo p in terms of Bernoulli numbers. As a consequence of Reinhart's recent counterexample to the Ankeny--Artin--Chowla conjecture, we find a prime p such that D_p^± ≡ 0 pmod p. This gives a negative answer to a question of the Sun.
ER  -