Find a Formula for Pi

​
minimal index in current formula
1
formula to choose
1
selectfirst equation
addnew equation
reset
Show 4
t
1
= π using the first 13 nonzero terms of the corresponding series.
current formula
chosen equations
t
746
-
t
1
+3
t
5
+
t
6
+
t
34
​
A Gregory number is a number
t
x
=arccot(x)
, where
x
is an integer or rational number. Expanding,
t
x
=
1
x
-
1
3
3
x
+
1
5
5
x
-…
. With
x=1
, we get Leibniz's formula for
π/4=1-
1
3
+
1
5
-…
, which converges too slowly. The larger
x
is, the better the approximation.
Euler found the formulas
t
1
=2
t
3
+
t
7
and
t
1
=5
t
7
+2
t
18
-2
t
57
and others. Machin calculated
π
to 100 decimals using
t
1
=4
t
5
-
t
239
.
Størmer's numbers
{1,2,4,5,6,9,10,11,12,…}
are the positive whole numbers
n
for which the largest prime factor of
2
n
+1
is at least
2n
. Størmer showed that every Gregory number
t
m
can be expressed uniquely as integer linear combinations of Gregory numbers with Störmer number indices.
Størmer found
t
1
=12
t
8
+8
t
57
-5
t
239
, and we found
t
1
=16
t
37
+4
t
44
+8
t
119
+4
t
179
-
t
239
+4
t
254
-4
t
882
. To find Størmer's formula, use
t
8
=
t
1
-
t
2
-
t
5
,
t
57
=-2
t
1
+3
t
2
+
t
5
,
t
239
=-
t
1
-4
t
5
, and eliminate
t
2
and
t
5
.
The Demonstration finds formulas for
t
1
where the terms on the right have indices as large as possible. Gregory numbers
t
n
are calculated for integers
n
that are not Størmer numbers
{3,7,8,13,17,…}
and formulas are selected according to the term of minimal index on the right side that is a Størmer number.
First, choose a formula with minimal index 1. Select the formula as the first equation. Pick out the next smallest index, say
m
, choose a formula in which
m
is minimal, and add that formula as a new equation; the Demonstration solves the system for
t
1
, eliminating the term with the index
m
. Now continue with the new smallest index on the right.

Details

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Copernicus Books/Springer, 2006 pp. 242–247.

External Links

Geometric Illustration of Machin-Like Formulas
Classical Approximations of Pi

Permanent Citation

Izidor Hafner
​
​"Find a Formula for Pi"​
​http://demonstrations.wolfram.com/FindAFormulaForPi/​
​Wolfram Demonstrations Project​
​Published: January 12, 2010