Why does this equation's solution almost equal pi?

[Hans de Vries]

Almost pi ...

What is the solution to:

$$\mbox{ \frac{1}{2}}\ \mbox{\Huge e}^{\ \frac{1}{2}X^2}\ -\ 2\ \mbox{\Huge e}^{\ 2X^{1/2}}\ =\ \left(\frac{1}{2} \right)^2}$$
Answer: x = 3.1415935362596164657060129064942...

Almost pi, the difference is only 2.8 10^-7.
Now, is this a coincidence or is there an explanation?
Regards, Hans

 

[lalbatros]

You should look at other examples to guess what the answer is.
Try solving:

f(x)-f(1/x) = (1/x)^x

for different functions f, and see how often youget near to pi.

 

[uart]

That's an interesting find hans. I'd say it's just coincidence but I'd be very interested to see any other ideas on this.

BTW. For anyone wanting to solve the equation for themselves and test han's claim, it is amendable to fix-point iteration as in :

x = sqrt( 2 log( 1/2 + 4 exp(2 sqrt(x) ) ) )

BTW. It's actually more like $$8.8 \times 10^{-7}$$ away from Pi, but it's still pretty close. :)

 

[waht]

One can say that this is just a first term of some infinite series or iteration for pi we don't know about.

 

[Gib Z]

Some amazing rapidly converging one too, but I doubt it. If a forum like us can realize this, surely mathematicians would have. And I know they haven't because this would prove pi and e are not algebraically independent, which is an open problem.