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Balkan MO
2001 Balkan MO
1
1
Part of
2001 Balkan MO
Problems
(1)
An easy diophantine-like equation: find the exponent of 2
Source: Balkan MO 2001, problem 1
4/24/2006
Let
a
,
b
,
n
a,b,n
a
,
b
,
n
be positive integers such that
2
n
−
1
=
a
b
2^n - 1 =ab
2
n
−
1
=
ab
. Let
k
∈
N
k \in \mathbb N
k
∈
N
such that
a
b
+
a
−
b
−
1
≡
0
(
m
o
d
2
k
)
ab+a-b-1 \equiv 0 \pmod {2^k}
ab
+
a
−
b
−
1
≡
0
(
mod
2
k
)
and
a
b
+
a
−
b
−
1
≠
0
(
m
o
d
2
k
+
1
)
ab+a-b-1 \neq 0 \pmod {2^{k+1}}
ab
+
a
−
b
−
1
=
0
(
mod
2
k
+
1
)
. Prove that
k
k
k
is even.
modular arithmetic
number theory unsolved
number theory