Regular "pentation"?

[andydude]

It would be interesting to determine what the upper bound of the regular interval for the bases of pentation would be. We know that for tetration it is \( e^{1/e} \), but what about pentation? (I suppose this would require tetration for b greater than \( e^{1/e} \) to investigate, though, so we'd need other methods like the Abel iteration or the continuum sum (I'm a big fan of continuum sums, by the way ))

I think I finally understand what you're talking about here. There is an interval that plays the same role in pentation as \( (e^{-e}, e^{1/e}) \) plays in tetration. I talked about the upper bound of this interval here, but whether or not there is a lower bound to this interval is unknown.

Just as the point \( (x=e^{1/e}, y=e) \) is the "highest" point on the graph of \( y = x {\uparrow}{\uparrow} \infty \),
so is \( (x=1.6353, y=3.0885) \) the "highest" point on the graph of \( y = x {\uparrow}{\uparrow}{\uparrow} \infty \).