Congruences for $q^{[p/8]}(mod p)$

• 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 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.

Kategorie tematyczne

• 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: 1-25

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/aa159-1-1