On the quadratic character of quadratic units

Abstract

Let p ≡ 1 ( mod 4 ) be a prime. Let a , b ∈ Z with p ∤ a ( a 2 + b 2 ) . In the paper we mainly determine ( b + a 2 + b 2 2 ) p − 1 2 ( mod p ) by assuming p = c 2 + d 2 or p = A x 2 + 2 B x y + C y 2 with A C − B 2 = a 2 + b 2 . As an application we obtain simple criteria for ε D to be a quadratic residue ( mod p ) , where D > 1 is a squarefree integer such that D is a quadratic residue of p, ε D 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 { U n } is the Lucas sequence defined by U 0 = 0 , U 1 = 1 and U n + 1 = b U n + k 2 U n − 1 ( n ⩾ 1 ) .