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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-1-1,
author = {Zhi-Hong Sun},
title = {Congruences for $q^{[p/8]}(mod p)$
},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {1-25},
zbl = {1292.11007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-1-1}
}
Zhi-Hong Sun. Congruences for $q^{[p/8]}(mod p)$
. Acta Arithmetica, Tome 161 (2013) pp. 1-25. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-1-1/