Problems and results on determinants involving Legendre symbols

Zhi-Wei Sun

Published 2024-05-06Version 1

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. After a review of known results, we present some new results and pose many conjectures for further research. For example, we conjecture that $$\det\left[x+\left(\frac{j-k}p\right)+\left(\frac jp\right)-\left(\frac kp\right)\right]_{0\le j,k\le (p-1)/2}=4$$ for any prime $p\equiv3\pmod4$, where $(\frac{\cdot}p)$ denotes the Legendre symbol.