Geometric Illustration of Machin-Like Formulas

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

.