[VA] SRC #016 - Pi Day 2024 Special

[J-F Garnier]

(03-14-2024 06:20 PM)Valentin Albillo Wrote:  Let x be the only real root of

      x³ - 6 x² + 4 x - 2 = 0

Compute ln(x²⁴ - 24) ÷ √H, where H is the number of hours in a week minus five.

The most amazing appearance !

Numerically solving the equation on Free42, I got a similar result as Gerson:

LBL "F7"
MVAR "X"
RCL "X"
ENTER
ENTER
ENTER
6
-
*
4
+
*
2
-
RTN

SOLVE "F7" for X between 1 and 9
then compute:
RCL "X" , 24 , Y^X , 24 , - , LN , 163 , SQRT , /
result = 3.141592653589793238462643383279504
within 1 ULP (1E-33) from the built-in PI constant !

Now, let's have a look at X:
X = 5.318628217750185659109680153318024
and X^24 = 262537412640768767.9999999999992527
is very close to an integer.
Also H = number of hours in a week minus five = 24*7-5 = 163.

This reminds us the famous approximation:
\(\pi\) ~ ln(640320^3+744) / √(163)
based on the so-called Ramanujan's constant.

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 !

J-F