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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.04642 |
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Table of Contents:
- 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 $φ(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $φ$ is Euler's totient function. This confirms a previous conjecture of the second author.