# Congruences for ${q}^{\left[p/8\right]}\left(modp\right)$

Acta Arithmetica (2013)

• Volume: 159, Issue: 1, page 1-25
• ISSN: 0065-1036

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## Abstract

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Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine ${q}^{\left[p/8\right]}\left(modp\right)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.

## How to cite

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Zhi-Hong Sun. "Congruences for $q^{[p/8]}(mod p)$." Acta Arithmetica 159.1 (2013): 1-25. <http://eudml.org/doc/279455>.

@article{Zhi2013,
abstract = {Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^\{[p/8]\}(mod p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.},
author = {Zhi-Hong Sun},
journal = {Acta Arithmetica},
keywords = {reciprocity law; octic residue; congruence; quartic Jacobi symbol},
language = {eng},
number = {1},
pages = {1-25},
title = {Congruences for $q^\{[p/8]\}(mod p)$},
url = {http://eudml.org/doc/279455},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Zhi-Hong Sun
TI - Congruences for $q^{[p/8]}(mod p)$
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 1
SP - 1
EP - 25
AB - Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p/8]}(mod p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.
LA - eng
KW - reciprocity law; octic residue; congruence; quartic Jacobi symbol
UR - http://eudml.org/doc/279455
ER -

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