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Putnam
1953 Putnam
B7
B7
Part of
1953 Putnam
Problems
(1)
Putnam 1953 B7
Source: Putnam 1953
7/16/2022
Let
w
∈
(
0
,
1
)
w\in (0,1)
w
∈
(
0
,
1
)
be an irrational number. Prove that
w
w
w
has a unique convergent expansion of the form
w
=
1
p
0
−
1
p
0
p
1
+
1
p
0
p
1
p
2
−
1
p
0
p
1
p
2
p
3
+
…
,
w= \frac{1}{p_0} - \frac{1}{p_0 p_1 } + \frac{1}{ p_0 p_1 p_2 } - \frac{1}{p_0 p_1 p_2 p_3 } +\ldots,
w
=
p
0
1
−
p
0
p
1
1
+
p
0
p
1
p
2
1
−
p
0
p
1
p
2
p
3
1
+
…
,
where
1
≤
p
0
<
p
1
<
p
2
<
…
1\leq p_0 < p_1 < p_2 <\ldots
1
≤
p
0
<
p
1
<
p
2
<
…
are integers. If
w
=
1
2
,
w= \frac{1}{\sqrt{2}},
w
=
2
1
,
find
p
0
,
p
1
,
p
2
.
p_0 , p_1 , p_2.
p
0
,
p
1
,
p
2
.
Putnam
irrational number
expansion