On the quartic character of quadratic units

• Autorzy: Zhi-Hong Sun
• Języki publikacji: EN

Abstrakty

EN

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, $d=2^r d₀$ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine $(b+√(b²+4^α)/2)^{(p-1)/4)} (mod p)$ for p = x²+(b²+4^{α})y² (b,x,y ∈ ℤ, 2∤b), and $(2a+√{4a²+1})^{(p-1)/4} (mod p)$ for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for $U_{(p-1)/4} (mod p)$ and the criterion for $p | U_{(p-1)/8}$ (if p ≡ 1 (mod 8)), where {Uₙ} is the Lucas sequence given by U₀ = 0, U₁ = 1 and $U_{n+1} = bUₙ+U_{n-1} (n≥1)$, and b ≢ 2 (mod 4). Hence we partially solve some conjectures that we posed in 2009.

Kategorie tematyczne

• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11A15: Power residues, reciprocity
• 11E25: Sums of squares and representations by other particular quadratic forms
• 11A07: Congruences; primitive roots; residue systems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 1
• Strony: 89-100

Daty

wydano

2013

Twórcy

autor

Zhi-Hong Sun

• School of Mathematical Sciences, Huaiyin Normal University, Huaian, Jiangsu 223001, P.R. China

Identyfikatory

DOI

10.4064/aa159-1-5