Cubic residues and binary quadratic forms

Abstract

Let p > 3 be a prime, u , v , d ∈ Z , gcd ( u , v ) = 1 , p ∤ u 2 − d v 2 and ( − 3 d p ) = 1 , where ( a p ) is the Legendre symbol. In the paper we mainly determine the value of ( u − v d u + v d ) ( p − ( p 3 ) ) / 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 ε d to be a cubic residue of p, where ε d is the fundamental unit of the quadratic field Q ( d ) . We also give a general criterion for p | U ( p − ( p 3 ) ) / 3 , where { U n } is the Lucas sequence defined by U 0 = 0 , U 1 = 1 and U n + 1 = P U n − Q U n − 1 ( n ⩾ 1 ). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.