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On determinants involving (frac{j²-k²}p) and (frac{jk}p)

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arxiv 2408.14401 v1 pith:N54FJSH4 submitted 2024-08-26 math.NT

On determinants involving (frac{j²-k²}p) and (frac{jk}p)

classification math.NT
keywords fracdeltaleftrightdeterminantcdotdependdeterminants
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Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{\delta\le j,k\le (p-1)/2}$$ with $\delta\in\{0,1\}$. For example, we prove that the determinant does not depend on $w$ if $p\equiv3\pmod4$ and $\delta=0$.

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