Natural Properties of the Tetra-Euler Number
#1
Question 
The tetra-Euler number is a "natural base" for tetration. (Denoted e4.) It is approximately equal to 3.089. (https://math.eretrandre.org/hyperops_wik...ler_number)
Other than the ones stated on the wiki page, what natural properties does it have?
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ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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#2
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.
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#3
(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.
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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#4
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?

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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#5
(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.
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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#6
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.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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#7
Question 
[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?
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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