Problems and results on determinants involving Legendre symbols

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ we have $$\det {[x+\left(\frac{j-k}{p}\right)+\left(\frac{j}{p}\right)-\left(\frac{k}{p}\right)]}_{0\le j,k\le (p-1)\mathrm{/}2}=4,$$ where $(\frac{\cdot }{p})$ is the Legendre symbol. We also pose many conjectures for further research. For example, for any prime $p>3$ we conjecture that \begin{align*}&\ \det\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac{jk}p\right)\right]_{1\le j,k\le(p-1)/2} \=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&$\text{if}$\ p\equiv1\pmod4, \(-1)^{(h(-p)-1)/2}(1-(2-(\frac 2p))h(-p))p^{(p-3)/4}&$\text{if}$\ p\equiv3\pmod4, \end{cases}\end{align*} where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$.