Cubic residues and binary quadratic forms Original Research Article

Let p>3 be a prime, image, gcd(u,v)=1, pdoes not divideu2−dv2 and image, where image is the Legendre symbol. In the paper we mainly determine the value of image by expressing p in terms of appropriate binary quadratic forms. As applications, for image we obtain a general criterion for image and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field image. We also give a general criterion for image, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=PUn−QUn−1 (ngreater-or-equal, slanted1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.