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PEN P Problems
18
18
Part of
PEN P Problems
Problems
(1)
P 18
Source:
5/25/2007
Let
p
p
p
be a prime with
p
≡
1
(
m
o
d
4
)
p \equiv 1 \pmod{4}
p
≡
1
(
mod
4
)
. Let
a
a
a
be the unique integer such that
p
=
a
2
+
b
2
,
a
≡
−
1
(
m
o
d
4
)
,
b
≡
0
(
m
o
d
2
)
p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}
p
=
a
2
+
b
2
,
a
≡
−
1
(
mod
4
)
,
b
≡
0
(
mod
2
)
Prove that
∑
i
=
0
p
−
1
(
i
3
+
6
i
2
+
i
p
)
=
2
(
2
p
)
,
\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),
i
=
0
∑
p
−
1
(
p
i
3
+
6
i
2
+
i
)
=
2
(
p
2
)
,
where
(
k
p
)
\left(\frac{k}{p}\right)
(
p
k
)
denotes the Legendre Symbol.
modular arithmetic
number theory
Additive Number Theory