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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 + (j-kp) + (jp) - (kp) ₀ ₉, ₊ (-₁) /₂=4 for any prime p34, where () denotes the Legendre symbol.
Zhi‐Wei Sun (Mon,) studied this question.