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Geometric Illustration of Machin-Like Formulas

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choose
Størmer decomposition of
t
m
where m =
s
i
th
i
decomposition of
t
1
i
1
t
1
t
2
+
t
3
This Demonstration illustrates geometrically some Machin-like formulas for computing the number
π
. A Gregory number is
t
x
=arcccot(x)
, where
x
is a rational number. Since
t
1
=π/4
, Gregory numbers arise in the determination of Machin-like formulas for computing the number
π
.
Størmer's numbers
s
i
=3,7,8,13,17,18,21,30,…
are the positive whole numbers
s
i
for which the largest prime factor of
2
s
i
+1
is at least
2
s
i
. Størmer showed that every Gregory number
t
m
, where
m
is a Störmer number, can be uniquely expressed as an integer linear combination of smaller
t
n
.
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