Problems and results on determinants involving Legendre symbols

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 [x+(j-kp)+( jp)-( kp)]0≤ j,k≤ (p-1)/2=4 for any prime p≡34, where (·p) denotes the Legendre symbol.