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National and Regional Contests
PEN Problems
PEN D Problems
23
23
Part of
PEN D Problems
Problems
(1)
D 23
Source:
5/25/2007
Let
p
p
p
be an odd prime of the form
p
=
4
n
+
1
p=4n+1
p
=
4
n
+
1
. [*] Show that
n
n
n
is a quadratic residue
(
m
o
d
p
)
\pmod{p}
(
mod
p
)
. [*] Calculate the value
n
n
n^{n}
n
n
(
m
o
d
p
)
\pmod{p}
(
mod
p
)
.
quadratics
modular arithmetic
Congruences