Find a Formula for Pi

A Gregory number is a number , where is an integer or rational number. Expanding, . With , we get Leibniz's formula for , which converges too slowly. The larger is, the better the approximation.
Euler found the formulas and and others. Machin calculated to 100 decimals using .
Størmer's numbers are the positive whole numbers for which the largest prime factor of is at least . Størmer showed that every Gregory number can be expressed uniquely as integer linear combinations of Gregory numbers with Störmer number indices.
Størmer found , and we found . To find Størmer's formula, use , , , and eliminate and .
The Demonstration finds formulas for where the terms on the right have indices as large as possible. Gregory numbers are calculated for integers that are not Størmer numbers 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 , choose a formula in which is minimal, and add that formula as a new equation; the Demonstration solves the system for , eliminating the term with the index . Now continue with the new smallest index on the right.

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DETAILS

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