Quartic residues and binary quadratic forms

✍ Scribed by Zhi-Hong Sun

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 426 KB
• Volume: 113
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.11.003

✦ Synopsis

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 d be the fundamental unit of the quadratic field Q(

Since 1942 many mathematicians tried to characterize those primes p so that d is a quadratic or quartic residue (mod p). In this paper we will completely solve these open problems by determining the value of

p ))/2 (mod p), where p is an odd prime, u, v, d ∈ Z, v = 0, gcd(u, v) = 1 and

As an application we also obtain a general criterion for p | u (p-( -1 p ))/4 (a, b), where {u n (a, b)} is the Lucas sequence defined by u 0 = 0, u 1 = 1 and u n+1 = bu n -au n-1 (n 1).