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 $p\equiv3\pmod4$ we have $$\det\left[x+\left(\frac{ j-k}p\right)+\left(\frac jp\right)-\left(\frac kp\right)\right]_{ 0\le j,k\le(p-1)/2}=4,$$ 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 $\mathbb Q(\sqrt{ -p})$.
Categories: math.NT