Quartic Residues and Binary Quadratic Forms

Let p ≡ 1 (mod 4) be a prime, m ∈ Z and p m. In this paper we obtain a general criterion for m to be a quartic residue (mod p) in terms of appropriate binary quadratic forms. Let d > 1 be a squarefree integer such that ( d p ) = 1, where ( d p ) is the Legendre symbol, and let εd be the fundamental unit of the quadratic field Q( √ d). Since 1942 many mathematicians tried to characterize those primes p so that εd is a quadratic or quartic residue (mod p). In this paper we will completely solve these open problems by determining the value of (u + v √ d) (p−(−1 p ))/2 (mod p), where p is an odd prime, u, v, d ∈ Z, v 6= 0, gcd(u, v) = 1 and (−d p ) = 1. As an application we also obtain a general criterion for p | u (p−(−1 p ))/4 (a, b), where {un(a, b)} is the Lucas sequence defined by u0 = 0, u1 = 1 and un+1 = bun − aun−1 (n ≥ 1). MSC: 11A15, 11E25, 11B39.