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National and Regional Contests
PEN Problems
PEN H Problems
50
50
Part of
PEN H Problems
Problems
(1)
H 50
Source:
5/25/2007
Show that the equation
y
2
=
x
3
+
2
a
3
−
3
b
2
y^{2}=x^{3}+2a^{3}-3b^2
y
2
=
x
3
+
2
a
3
−
3
b
2
has no solution in integers if
a
b
≠
0
ab \neq 0
ab
=
0
,
a
≢
1
(
m
o
d
3
)
a \not\equiv 1 \; \pmod{3}
a
≡
1
(
mod
3
)
,
3
3
3
does not divide
b
b
b
,
a
a
a
is odd if
b
b
b
is even, and
p
=
t
2
+
27
u
2
p=t^2 +27u^2
p
=
t
2
+
27
u
2
has a solution in integers
t
,
u
t,u
t
,
u
if
p
∣
a
p \vert a
p
∣
a
and
p
≡
1
(
m
o
d
3
)
p \equiv 1 \; \pmod{3}
p
≡
1
(
mod
3
)
.
modular arithmetic
Diophantine Equations