More on e^(pi*sqrt(163))

[tpi...@gmail.com]

Hello all,

It is quite well-known that:

e^(pi*sqrt(19)) ~ 96^3 + 744
e^(pi*sqrt(43)) ~ 960^3 + 744
e^(pi*sqrt(67)) ~ 5280^3 + 744
e^(pi*sqrt(163)) ~ 640320^3 + 744

using the four highest Heegner numbers. But it is not so well-known
that the expression e^(pi*sqrt(d)) can be given *another* internal
structure:

e^(pi*sqrt(19)) ~ 12^3(3^2-1)^3 + 744
e^(pi*sqrt(43)) ~ 12^3(9^2-1)^3 + 744
e^(pi*sqrt(67)) ~ 12^3(21^2-1)^3 + 744
e^(pi*sqrt(163)) ~ 12^3(231^2-1)^3 + 744

The reason for the squares are due to certain Eisenstein series -- but
that's another story. :-)

Beautifully consistent, aren't they?

I'm working on a new webpage about this and, er, other Ramanujan-
related stuff. But I'm having a devil of a time finishing it due to my
day job. I'll post the link here when it's done.

Yours,

Titus