On the quadratic character of quadratic units Original Research Article

Let image be a prime. Let image with pdoes not dividea(a2+b2). In the paper we mainly determine image by assuming p=c2+d2 or p=Ax2+2Bxy+Cy2 with AC−B2=a2+b2. As an application we obtain simple criteria for εD to be a quadratic residue image, where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field image with negative norm. We also establish the congruences for image and obtain a general criterion for pU(p−1)/4, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=bUn+k2Un−1 (ngreater-or-equal, slanted1).