Quadratic versions of these series correspond to class number two imaginary quadratic fields. The most spectacular and largest example has d = -427 and
For a further discussion of these see [2] where many such quadratic
examples are considered. Various of the recent record setting calculations
of pi have been based on these series. In particular the Chudnovskys
computed over two billion digits of using the second series above.
There is an unlimited number of such series with increasingly more rapid
convergence. The price one pays is that one must deal with more
complicated algebraic irrationalities. Thus a class number p field will
involve degree algebraic integers as the constants
,
and
in the series. The largest class number three example
of
corresponds to d = -907 and gives 37 or 38 digits per
term. It is
The Computational Component. The absolute invariant,
and so the coefficients A, B, and C satisfy polynomial equations of
known degree and height. Thus the problem of determining the coefficients
of each series reduces to algebra and can be entirely automated. This is
really the dream case for computer aided analysis. Indeed from the
expressions for ,
,
we straightforwardly computed their
values to several hundred digits. The lattice basis reduction algorithm, as
implemented in Maple, now provides the minimal polynomials for each
quantity. In addition, a higher precision calculation actually provides a
proof of the claimed identity. This last step requires knowing a priori
bounds on the degrees and heights of the invariants. While somewhat
mathematically sophisticated, the computation required is fairly easy
though a little slow.