MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN H Problems
30
30
Part of
PEN H Problems
Problems
(1)
H 30
Source:
5/25/2007
Let
a
a
a
,
b
b
b
,
c
c
c
be given integers,
a
>
0
a>0
a
>
0
,
a
c
−
b
2
=
p
ac-b^2=p
a
c
−
b
2
=
p
a squarefree positive integer. Let
M
(
n
)
M(n)
M
(
n
)
denote the number of pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
for which
a
x
2
+
b
x
y
+
c
y
2
=
n
ax^2 +bxy+cy^2=n
a
x
2
+
b
x
y
+
c
y
2
=
n
. Prove that
M
(
n
)
M(n)
M
(
n
)
is finite and
M
(
n
)
=
M
(
p
k
⋅
n
)
M(n)=M(p^{k} \cdot n)
M
(
n
)
=
M
(
p
k
⋅
n
)
for every integer
k
≥
0
k \ge 0
k
≥
0
.
Diophantine Equations