Is the number $1.201~943\dots$ of any significance?

Question

I ran the function $(x^2+y^2)^z=z$, where $z$ is a constant. The function produced a circle, and the $z$ value where the radius of the circle turned out to be the largest was $e$. At that point, the radius of said circle was $1.201~943\dots$ Since it correlates to $e$ in that way, I wonder of $1.201~943\dots$ has any significance, or a special name. If it does, I could potentially use it somehow. Is this number important?

Answer 1

This number is just $e^{\frac1{2e}}$. Nothing especially noteworthy about it as far as I know.

Note your equation can be written as $$r=z^{\frac1{2z}}$$ and you have maximized this.

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Addendum: Note that $$\frac{dr}{dz}=\tfrac12 z^{(2z)^{-1}-2}(1-\ln z)$$ The sign of this expression is determined by $1-\ln z$, which shows the function $r(z)$ is increasing to the left of $z=e$ and decreasing to the right of $z=e$. The function is therefore maximized at $z=e$.

Answer 2

Your equation is $$x^2+y^2= \left(z^{\frac{1}{2z}}\right)^2$$

The function $z^{\frac{1}{2z}}$ attains its maximum when $z=e$ and the value is $e^{\frac{1}{2e}}$, your number must be equal to $e^{\frac{1}{2e}}$.