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Balkan MO
1993 Balkan MO
4
4
Part of
1993 Balkan MO
Problems
(1)
Equation has non-trivial solution iff $m=p$ (Balkan 1993)
Source: Balkan MO 1993, Problem 4
4/25/2006
Let
p
p
p
be a prime and
m
≥
2
m \geq 2
m
≥
2
be an integer. Prove that the equation
x
p
+
y
p
2
=
(
x
+
y
2
)
m
\frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m
2
x
p
+
y
p
=
(
2
x
+
y
)
m
has a positive integer solution
(
x
,
y
)
≠
(
1
,
1
)
(x, y) \neq (1, 1)
(
x
,
y
)
=
(
1
,
1
)
if and only if
m
=
p
m = p
m
=
p
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