Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .