Very accurate pi approximation

Yesterday I ran across this approximation for and posted it on X.

3 log(640320) / 163

This approximation is good to 15 decimal places, and so the approximation is exact to within the limits of floating point arithmetic.

I said in a follow up comment that the 163 term looked familiar from Heegner numbers and near integers. I was so close but didn't make the full connection.

It's well known that exp(163) is very nearly an integer N:

exp(163) =N -

whereN = 262537412640768744 and is less than 10^-12. The left-hand side of this equation is known as Ramanujan's constant.

Incidentally, Martin Gardner announced on April Fool's day in 1975 that Ramanujan's constant had been proven to be an integer.

There are deep reasons for why Ramanujan's constant is nearly an integer, but the connection between this fact and the approximation for is not deep at all: the former is basically the latter, solved for . Because

N = 640320^3 + 744,

log(N) is very nearly 3 log(640320).

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