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

Acta Arithmetica (2013)

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

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topZhi-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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