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Natural Properties of the Tetra-Euler Number - Printable Version

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Natural Properties of the Tetra-Euler Number - Catullus - 06/11/2022

The tetra-Euler number is a "natural base" for tetration. (Denoted e4.) It is approximately equal to 3.089. (https://math.eretrandre.org/hyperops_wiki/index.php?title=Euler_number)
Other than the ones stated on the wiki page, what natural properties does it have?

RE: Natural Properties of the Tetra-Euler Number - JmsNxn - 06/11/2022

The way you can think of this base value, as the next \(\eta\). It's actually pretty ugly for tetration. It will involve repelling branches, which when iterated, produce bounded pentations. So \(\eta^4\) is to \(\uparrow^3\) as, \(\eta\) is to \(\uparrow^2\). This becomes much more complicated though, be cause \(\eta^4\) is defined off of a repelling iteration. It does not exist using solely attracting iterations of tetration.

RE: Natural Properties of the Tetra-Euler Number - Catullus - 06/11/2022

(06/11/2022, 04:57 AM)JmsNxn Wrote: The way you can think of this base value, as the next \(\eta\).
No. The Tetra-Euler Number is not the next eta. The tetra-critical base is the next eta. It is about 1.635.

RE: Natural Properties of the Tetra-Euler Number - MphLee - 06/15/2022

A new general definition of the eulers and the etas should be added to the wiki imho.
Question: I don't remember the literature atm. Was this definition already given been given in some article/paper?

RE: Natural Properties of the Tetra-Euler Number - Catullus - 06/15/2022

(06/15/2022, 11:02 PM)MphLee Wrote: A new general definition of the eulers and the etas should be added to the wiki imho.
Question: I don't remember the literature atm. Was this definition already given been given in some article/paper?
What should the new definition be?
It was based off original research.

RE: Natural Properties of the Tetra-Euler Number - MphLee - 06/15/2022

When I've time I'll give a try.
I'm thinking about defining th whole sequences of etas and eulers as a function from ranks into something, atm idk what, that satisfies by definition the characterization given on the wiki.

RE: Natural Properties of the Tetra-Euler Number - Catullus - 07/01/2022

[Image: png.image?\dpi%7B110%7D%20e=\lim_%7Bx\to\infty...parrow%20x]. Is there a similar limit for the tetra-Euler number?
[Image: png.image?\dpi%7B110%7D%20e%5Ex] is its own derivative. Is there a derivative-like operation, such that when applied to [Image: png.image?\dpi%7B110%7De_4\uparrow\uparrow%20x] outputs [Image: png.image?\dpi%7B110%7De_4\uparrow\uparrow%20x]?
If so what is is?