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National and Regional Contests
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PEN H Problems
49
H 49
H 49
Source:
May 25, 2007
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group theory
abstract algebra
Diophantine Equations
Problem Statement
Show that the only solutions of the equation
x
3
−
3
x
y
2
−
y
3
=
1
x^{3}-3xy^2 -y^3 =1
x
3
−
3
x
y
2
−
y
3
=
1
are given by
(
x
,
y
)
=
(
1
,
0
)
,
(
0
,
−
1
)
,
(
−
1
,
1
)
,
(
1
,
−
3
)
,
(
−
3
,
2
)
,
(
2
,
1
)
(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)
(
x
,
y
)
=
(
1
,
0
)
,
(
0
,
−
1
)
,
(
−
1
,
1
)
,
(
1
,
−
3
)
,
(
−
3
,
2
)
,
(
2
,
1
)
.
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