Very accurate pi approximation

Very accurate pi approximation

Yesterday I ran across this approximation for π and posted it on X. π ≈ 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(π√163) is very nearly an integer N: exp(…