Quartic, octic residues and Lucas sequences

Abstract: Let be a prime and with . Suppose 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 is the Lucas sequence given by , and . We also pose many conjectures concerning , or . [Copyright &y& Elsevier]