On certain determinants and the square root of some determinants involving Legendre Symbols

Let $p>3$ be a prime and (./p) be the Legendre symbol. For any integer d with p d and any positive integer m, Sun introduced the determinants Tₘ(d,p)=[(i²+dj²)ᵐ(i²+dj²/p)]1 i,j (p-1)/2, and Dₚ⁽ᵐ⁾= [(i²-j²)ᵐ(i²-j²/p)]1≤ i,j≤ (p-1)/2 . In this paper, we obtain some properties of Tₘ (d,p) and √Dₚ⁽ᵐ⁾ for some m. We also confirm some related conjectures posed by Zhi-Wei Sun.