This function, j(tau), is responsible for some amazing approximations. For example, let,
k = 5280(236674+30303*Sqrt[61])
then,
e^(Pi*Sqrt(427)) = k^3 + 743.999999999999999999999987...
which has 22 consecutive nines. (Note that 427 = 7*61.) Then of course there's the famous d = -163 where k is an integer. It turns out that even the formulas for j(tau) are interesting in their own right. They can be of the form j(tau) = f(x)/g(x) for some polynomials in a function x and by defining the expression,
S = -Numerator + 1728Denominator
then S is either a square or a near-square. Now why is that? Furthermore, there are also formulas in terms of continued fractions.
