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PEN P Problems
17
P 17
P 17
Source:
May 25, 2007
quadratics
Additive Number Theory
pen
Problem Statement
Let
p
p
p
be a prime number of the form
4
k
+
1
4k+1
4
k
+
1
. Suppose that
r
r
r
is a quadratic residue of
p
p
p
and that
s
s
s
is a quadratic nonresidue of
p
p
p
. Show that
p
=
a
2
+
b
2
p=a^{2}+b^{2}
p
=
a
2
+
b
2
, where
a
=
1
2
∑
i
=
1
p
−
1
(
i
(
i
2
−
r
)
p
)
,
b
=
1
2
∑
i
=
1
p
−
1
(
i
(
i
2
−
s
)
p
)
.
a=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-r)}{p}\right), b=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-s)}{p}\right).
a
=
2
1
i
=
1
∑
p
−
1
(
p
i
(
i
2
−
r
)
)
,
b
=
2
1
i
=
1
∑
p
−
1
(
p
i
(
i
2
−
s
)
)
.
Here,
(
k
p
)
\left( \frac{k}{p}\right)
(
p
k
)
denotes the Legendre Symbol.
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