Arithmetic properties of some permanents

In this paper we study arithmetic properties of some permanents, many of which involve trigonometric functions. For any primitive $n$-th root $\zeta$ of unity, we obtain closed formulas for the permanents $$\mathrm{per}\left[1-\zeta^jx_k\right]_{1\le j,k\le n}\ \ \text{and}\ \ \mathrm{per}\left[\frac1{1-\zeta^{j-k}x}\right]_{1\le j,k\le n}.$$ Another typical result states that for any odd integer $n>1$ we have $$t_n:=\frac1{\sqrt n}\mathrm{per}\left[\tan\pi\frac{jk}n\right]_{1\le j,k\le (n-1)/2}\in\mathbb Z,$$ and that $t_p\equiv(-1)^{(p+1)/2}\pmod p$ for any odd prime $p$. We also pose several conjectures for further research; for example, we conjecture that $$\mathrm{per}[|j-k|]_{1\le j,k\le p}\equiv-\frac12\pmod p$$ for any odd prime $p$.