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PEN H Problems
39
H 39
H 39
Source:
May 25, 2007
quadratics
Diophantine Equations
Problem Statement
Let
A
,
B
,
C
,
D
,
E
A, B, C, D, E
A
,
B
,
C
,
D
,
E
be integers,
B
≠
0
B \neq 0
B
=
0
and
F
=
A
D
2
−
B
C
D
+
B
2
E
≠
0
F=AD^{2}-BCD+B^{2}E \neq 0
F
=
A
D
2
−
BC
D
+
B
2
E
=
0
. Prove that the number
N
N
N
of pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
A
x
2
+
B
x
y
+
C
x
+
D
y
+
E
=
0
,
Ax^{2}+Bxy+Cx+Dy+E=0,
A
x
2
+
B
x
y
+
C
x
+
Dy
+
E
=
0
,
satisfies
N
≤
2
d
(
∣
F
∣
)
N \le 2 d( \vert F \vert )
N
≤
2
d
(
∣
F
∣
)
, where
d
(
n
)
d(n)
d
(
n
)
denotes the number of positive divisors of positive integer
n
n
n
.
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