On the Quadratic Character of Quadratic Units

Let p ≡ 1 (mod 4) be a prime. Let a, b ∈ Z with p a(a2 + b2). In the paper we mainly determine ( b+ √ a2+b2 2 ) p−1 2 (mod p) 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 (mod p), 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 Q( √ D) with negative norm. We also establish the congruences for U(p±1)/2 (mod p) and obtain a general criterion for p | U(p−1)/4, where {Un} is the Lucas sequence defined by U0 = 0, U1 = 1 and Un+1 = bUn + kUn−1 (n ≥ 1). MSC: Primary 11A15, Secondary 11B39, 11A07, 11E25, 11E16