Legendre Symbols Related to Certain Determinants.

Let p be an odd prime. For b , c ∈ Z , Sun introduced the determinant D p (b , c) = (i 2 + b i j + c j 2) p - 2 1 ⩽ i , j ⩽ p - 1 and investigated the Legendre symbol ( D p (b , c) p) . Recently Wu, She and Ni proved that ( D p (1 , 1) p) = (- 2 p) if p ≡ 2 (mod 3) , which confirms a previous conjecture of Sun. In this paper, we determine ( D p (1 , 1) p) in the case p ≡ 1 (mod 3) . Sun proved that D p (2 , 2) ≡ 0 (mod p) if p ≡ 3 (mod 4) ; in contrast, we prove that ( D p (2 , 2) p) = 1 if p ≡ 1 (mod 8) , and ( D p (2 , 2) p) = 0 if p ≡ 5 (mod 8) . Our tools include generalized trinomial coefficients and Lucas sequences. [ABSTRACT FROM AUTHOR]