Natural Properties of the Tetra-Euler Number Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022   06/11/2022, 04:18 AM (This post was last modified: 08/14/2022, 09:21 PM by Catullus.) 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? Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ JmsNxn Ultimate Fellow Posts: 985 Threads: 117 Joined: Dec 2010 06/11/2022, 04:57 AM 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. Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022 06/11/2022, 05:00 AM (This post was last modified: 08/19/2022, 10:28 AM by Catullus.) (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\ MphLee Long Time Fellow Posts: 341 Threads: 27 Joined: May 2013 06/15/2022, 11:02 PM 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 $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022 06/15/2022, 11:04 PM (This post was last modified: 08/14/2022, 09:23 PM by Catullus.) (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\ MphLee Long Time Fellow Posts: 341 Threads: 27 Joined: May 2013 06/15/2022, 11:07 PM 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 $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022   07/01/2022, 08:16 AM (This post was last modified: 08/19/2022, 10:29 AM by Catullus.) $\dpi{110} e=\lim_{x\to\infty}(1+1\div x)\uparrow x$. Is there a similar limit for the tetra-Euler number? $\dpi{110} e^x$ is its own derivative. Is there a derivative-like operation, such that when applied to $\dpi{110}e_4\uparrow\uparrow x$ outputs $\dpi{110}e_4\uparrow\uparrow x$? If so what is is? Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ « Next Oldest | Next Newest »

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