Cubic Residues and Binary Quadratic Forms

Let p > 3 be a prime, u, v, d ∈ Z, gcd(u, v) = 1, p u2 − dv2 and (−3d p ) = 1, where ( p ) is the Legendre symbol. In the paper we mainly determine the value of u−v √ d u+v √ d (p−( p3 ))/3 (mod p) by expressing p in terms of appropriate binary quadratic forms. As applications, for p ≡ 1 (mod 3) we obtain a general criterion for m(p−1)/3 (mod p) and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field Q(d). We also give a general criterion for p | U(p−( p3 ))/3, where {Un} is the Lucas sequence defined by U0 = 0, U1 = 1 and Un+1 = PUn − QUn−1 (n ≥ 1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms. MSC: Primary 11A15, Secondary 11E16, 11A07, 11B39