Cubic residues and binary quadratic forms

AbstractLet p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p∤u2−dv2 and (−3dp)=1, where (ap) is the Legendre symbol. In the paper we mainly determine the value of (u−vdu+vd)(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