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Putnam
1959 Putnam
B6
B6
Part of
1959 Putnam
Problems
(1)
Prove this.
Source:
5/2/2020
Let
α
\alpha
α
and
β
\beta
β
be irrational numbers with the property that
1
α
+
1
β
=
1
\frac{1}{\alpha} +\frac{1}{\beta}=1
α
1
+
β
1
=
1
Let
{
a
n
}
\{a_n\}
{
a
n
}
and
{
b
n
}
\{b_n\}
{
b
n
}
be the sequences given by
a
n
=
⌊
n
α
⌋
a_n= \lfloor n\alpha \rfloor
a
n
=
⌊
n
α
⌋
and
b
n
=
⌊
n
β
⌋
b_n= \lfloor n\beta \rfloor
b
n
=
⌊
n
β
⌋
respectively. Prove that the sequences
{
a
n
}
\{ a_n\}
{
a
n
}
and
{
b
n
}
\{ b_n \}
{
b
n
}
has no term in common and cover all the natural numbers. I know this theorem from long ago, but forgot the proof of it. Can anybody help me with this?
algebra
number theory