A Gregory number is a number
is an integer or rational number. Expanding,
, we get Leibniz's formula for
, which converges too slowly. The larger
is, the better the approximation.
Euler found the formulas
and others. Machin calculated
to 100 decimals using
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.
, and we found
. To find Størmer's formula, use
, and eliminate
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
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.