In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime p > 3 with p ≡ 3 (mod 4) and a, b ∈ ℤ with p ∤ ab, we prove that {1 + \, {a{j² + b{k²}}p} ₁ ₉, ₊ - ₁{2}} = \ {array{*{20c} - {2^{{p - 12}}p^{{p - 34}}, }&if ({{abp}) = 1, } \\ {p^{{p - 34}}, }&if ({{abp}) = - 1, } array}. denotes the Legendre symbol. We also pose some conjectures for further research.
Zhi-Wei Sun (Tue,) studied this question.
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