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4
Look at the exponent...
Look at the exponent...
Source:
October 21, 2010
number theory unsolved
number theory
Problem Statement
Prove that the following statement is true for two natural nos.
m
,
n
m,n
m
,
n
if and only
v
(
m
)
=
v
(
n
)
v(m) = v(n)
v
(
m
)
=
v
(
n
)
where
v
(
k
)
v(k)
v
(
k
)
is the highest power of
2
2
2
dividing
k
k
k
.
∃
\exists
∃
a set
A
A
A
of positive integers such that
(
i
)
(i)
(
i
)
x
,
y
∈
N
,
∣
x
−
y
∣
=
m
⟹
x
∈
A
x,y \in \mathbb{N}, |x-y| = m \implies x \in A
x
,
y
∈
N
,
∣
x
−
y
∣
=
m
⟹
x
∈
A
or
y
∈
A
y \in A
y
∈
A
(
i
i
)
(ii)
(
ii
)
x
,
y
∈
N
,
∣
x
−
y
∣
=
n
⟹
x
∉
A
x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A
x
,
y
∈
N
,
∣
x
−
y
∣
=
n
⟹
x
∈
A
or
y
∉
A
y \not\in A
y
∈
A
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