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.