On the quartic character of quadratic units

Zhi-Hong Sun

Acta Arithmetica, Tome 161 (2013), p. 89-100 / Harvested from The Polish Digital Mathematics Library

Résumé

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+4α)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.

Détail

Publié le : 2013-01-01

Zbl 1287.11005

EUDML-ID : urn:eudml:doc:279410