Some new results on determinants and permanents

In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ the number $2\det [a_{jk}{]}_{0\le j,k\le (p-1)\mathrm{/}2}$ 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 $\mathrm{p}\mathrm{e}\mathrm{r}[j^{k-1}{]}_{1\le j,k\le n-1}\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}n)$ for any integer $n>1$ with $n\equiv ̸2(\mathrm{m}\mathrm{o}\mathrm{d}4)$.