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arXiv · 2408.07034

On a determinant involving linear combinations of Legendre symbols

Keqin Liu·Zhi-Wei Sun·Li-Yuan Wang
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Abstract

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

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Keqin Liu, Zhi-Wei Sun, Li-Yuan Wang. 2024-08-13. On a determinant involving linear combinations of Legendre symbols. https://arxiv.org/abs/2408.07034

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