On a determinant involving linear combinations of Legendre symbols

2024/08/13 by Keqin Liu, Zhi-Wei Sun, Zhi‐Wei Sun +4
Physics and Astronomy · #Advanced Mathematical Theories and Applications #math.NT

paper · pdf · doi:10.48550/arxiv.2408.07034

openalex publication_date 2024/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove a conjecture of the second author by evaluating the determinant det[x + (\fraci-jp) + (\frac ip)y + (\frac jp)z + (\fracijp)w]_{0≤ i,j≤(p-3)/2} for any odd prime p, where (\frac⋅p) denotes the Legendre symbol. In particular, the determinant is equal to x when p≡ 3\pmod4.

Citations

• On the cyclotomic field Q(e2πi/p) and Zhi-Wei Sun's conjecture on det M

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