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PEN C Problems
6
C 6
C 6
Source:
May 25, 2007
algebra
polynomial
absolute value
Quadratic Residues
pen
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be integers and let
p
p
p
be an odd prime with
p
∤
a
and
p
∤
b
2
−
4
a
c
.
p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.
p
∣
a
and
p
∣
b
2
−
4
a
c
.
Show that
∑
k
=
1
p
(
a
k
2
+
b
k
+
c
p
)
=
−
(
a
p
)
.
\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).
k
=
1
∑
p
(
p
a
k
2
+
bk
+
c
)
=
−
(
p
a
)
.
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