On the quartic character of quadratic units

Zhi-Hong Sun

Acta Arithmetica (2013)

  • Volume: 159, Issue: 1, page 89-100
  • ISSN: 0065-1036

Abstract

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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 d ₀ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine ( b + √ ( b ² + 4 α ) / 2 ) ( p - 1 ) / 4 ) ( m o d p ) for p = x²+(b²+4α)y² (b,x,y ∈ ℤ, 2∤b), and ( 2 a + √ 4 a ² + 1 ) ( p - 1 ) / 4 ( m o d 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 ( m o d 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 = b U ₙ + U n - 1 ( n ≥ 1 ) , and b ≢ 2 (mod 4). Hence we partially solve some conjectures that we posed in 2009.

How to cite

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Zhi-Hong Sun. "On the quartic character of quadratic units." Acta Arithmetica 159.1 (2013): 89-100. <http://eudml.org/doc/279410>.

@article{Zhi2013,
abstract = {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.},
author = {Zhi-Hong Sun},
journal = {Acta Arithmetica},
keywords = {congruence; quartic residue; Lucas sequence},
language = {eng},
number = {1},
pages = {89-100},
title = {On the quartic character of quadratic units},
url = {http://eudml.org/doc/279410},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Zhi-Hong Sun
TI - On the quartic character of quadratic units
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 1
SP - 89
EP - 100
AB - 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.
LA - eng
KW - congruence; quartic residue; Lucas sequence
UR - http://eudml.org/doc/279410
ER -

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