[VA] SRC #016 - Pi Day 2024 Special

[Gerson W. Barbosa]

(03-20-2024 08:16 AM)J-F Garnier Wrote:  Here the approximation is even better because the value X^24-24 itself is an excellent approximation of this Ramanujan's constant, coming from the root of a very simple 3rd-degree polynomial. Amazing !

If a tiny bit involving 640320² and a few other perfect squares is added to the second 24 constant then the approximation gets even more amazing:

log((2+(5+1/3√(163/3))^(1/3)+(5-1/3√(163/3))^(1/3))^24-(24+1/((32^2+36^2)*640320^2+(41^2+105^2)/69^2)))/√163

or

\[\frac{\ln\left[{\left({2+\sqrt[3]{5+\frac{1}{3}\sqrt{\frac{163}{3}}}+\sqrt[3]{5-\frac{1}{3}\sqrt{\frac{163}{3}}}}\right)^{24}-\left({24+\frac{1}{\left({32^2+36^2}\right)\times640320^2+\frac{41^2+105^2}{69^2​}}}\right)}\right]}{\sqrt{163}}\]


But then Free42 ought to be twice as precise in order to properly evaluate it.

Gerson.