Bummer!

[bo198214]

x of max |y| may be 3.08853227...(we remember that this new number succeed "Euler")

Nuninho, you'll have to explain a little more before I can understand. The graph is the difference between the two superfunctions of sqrt(2), which are nearly identical going from f(x)=4 downto f(x)=2.

I think I understood now what he meant!
Like the Euler number is the argument x where b^x = x for the b where b^x has exactly one fixpoint - in the transition from two fixpoints to no real fixpoint (i.e. \( b=e^{1/e} \)),
the "succeeding Euler number" is the argument x where b[4]x = x for the b where b[4]x has exactly one fixpoint - in the transition from two fixpoints to no real fixpoint (on x>0, because there is always a fixpoint between -2 and 0 for the tetrational.)
This b is around 1.635....

And it looks in the graph as if the maximum of the difference is achieved exactly at this new Euler number.

PS: We also have the analogon to the function \( x^{1/x} \), this function has its maximum at x=e and the maximum is e^(1/e). Same with the self-tetra-root it has its maximum at the "succeeding Euler number" 3.08853227... and its value is around 1.635... as Mike found out in this thread.

PS2: Ya and it also has the analogon to the maximum b such that b[4]oo exists (which is e^(1/e)[4]oo = e). So the maximum base for which b[5]oo exists is \( 1.635[5]\infty\approx 3.0885 \). Andrew/Nuninho pointed it out in this thread.