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arXiv · 2407.04642

Some determinants involving binary forms

Yue-Feng She·Zhi-Wei Sun
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Abstract

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author.

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Yue-Feng She, Zhi-Wei Sun. 2024-07-05. Some determinants involving binary forms. https://arxiv.org/abs/2407.04642

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