Some new results on determinants and permanents

Abstract

In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\equiv 3(\mathrm{mod}4)$ the number $2\det [a_{jk}{]}_{0\le j,k\le (p-1)/2}$ is congruent to a square modulo $p$, where $a_{jk}=(\frac{j+k}{p})+(\frac{j^2+k^2}{p})$ with $(\frac{\cdot }{p})$ the Legendre symbol. We also prove that $\mathrm{per}[j^{k-1}{]}_{1\le j,k\le n-1}\equiv 0(\mathrm{mod}n)$ for any integer $n>1$ with $n\not\equiv 2(\mathrm{mod}4)$.

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