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Journal Article
A NEW CHARACTERISATION FOR QUARTIC RESIDUACITY OF $\\mathbf {2}
2022
Overview
Let p be a prime with
$p\\equiv 1\\pmod {4}$
. Gauss first proved that
$2$
is a quartic residue modulo p if and only if
$p=x^2+64y^2$
for some
$x,y\\in \\Bbb Z$
and various expressions for the quartic residue symbol
$(\\frac {2}{p})_4$
are known. We give a new characterisation via a permutation, the sign of which is determined by
$(\\frac {2}{p})_4$
. The permutation is induced by the rule
$x \\mapsto y-x$
on the
$(p-1)/4$
solutions
$(x,y)$
to
$x^2+y^2\\equiv 0 \\pmod {p}$
satisfying
$1\\leq x < y \\leq (p-1)/2$
.
Publisher
Cambridge University Press
Subject
