Hyper-Operational Salad Numbers Catullus Fellow   Posts: 213 Threads: 47 Joined: Jun 2022 07/10/2022, 03:35 AM (This post was last modified: 07/12/2022, 11:07 AM by Catullus.) Can you please tell me some salad numbers that use hyper-operations? If you want, you may make up your own hyper-operational salad numbers and post them here. A salad number is a large number created with a mishmash of numbers or functions. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ Catullus Fellow   Posts: 213 Threads: 47 Joined: Jun 2022 08/05/2022, 03:57 AM (This post was last modified: 12/20/2022, 10:44 PM by Catullus.) Please let $$f(x)$$ equal $$(9\def\t{\uparrow}\t\t\t x)\def\e{\Lambda}\e^{\t\t\t}$$, where $$x\e^{\def\{{\underbrace{\t\t\cdots\t\t}}\{_y}$$ is defined for $$y\in\Bbb N$$ as the Bouncing Factorial of x, but instead of starting at 1, you start at 2, and you replace the multiplication in the definition with (y+2)-ation. Please let $$g(x)$$ equal $$f^{x\e^{\{_{f^{f^x(x)}x}}}(x)$$. Please let $$\def\\{c_\def\({\alpha}\(}\\$$ be the Catullus hierarchy with these rules: $$c_0(t)=$$ $$\phi$$$$(g^{g^t(t)}(t),g^{g^{g^{g^{t\e^{\{_{g^{g^{g^t(t)}(t)}(t)}}}(t)}(t)}(t)}(t),\def\ {\omega}\ ^{\ ^{\ ^9}})$$ $$c_{\(+1}(t)=\phi(\\^{\\^{\\^t(t)}(t)}(t),\\^{\\^{\\(t)\t^{\\^{\\^{\\^{t\e^{\{_{\\^{\\^{\\^{\\^{t\e^\t}(t)}(t)}(t)}(t)}}(t)}(t)}(t)}}\\^t(t)}(t)}(t),\ ^{\ ^{\ ^{\ ^\(}}})$$ $$\\(t)=c_{\([t]}(t)\iff\(\in\text{Lim}$$ I define hyper bouncing guppy as $$c_{\def\}{\varepsilon_0}\}}($$$$\text{guppy}$$$$)+1$$ with respect to the Wainer Hierachy system of fundamental sequences and $$\}[n]={}^n\$$. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ Daniel Fellow   Posts: 248 Threads: 80 Joined: Aug 2007 08/07/2022, 06:02 AM (This post was last modified: 08/07/2022, 07:00 AM by Daniel.) (08/05/2022, 03:57 AM)Catullus Wrote: Please let  equal , where  is defined for  as the Bouncing Factorial of x, but instead of starting at one you start at two and you replace the multiplication in the definition with (y+2)-ation. Please let  equal . Please let  equal . I define my number as h^Guppyplex(Guppyplex)+1. Wow Catullus, your dedication to nice typesetting is impressive. In my experience many problems in physics and mathematics have answers because they mean something. Even more, the best problems are at the perimeter of our understanding. Asking questions is great, but some of your questions are far beyond the limit of our collective progress. Understanding the motivation behind a problem is usually the first step in solving it. Paul Erdős talked about "The Book", of the ultimate mathematics theorems. This is a good guide for what to work on. Sorry, I have no idea as to how to answer your question. Daniel JmsNxn Ultimate Fellow     Posts: 1,178 Threads: 123 Joined: Dec 2010 08/07/2022, 08:44 AM (08/07/2022, 06:02 AM)Daniel Wrote: (08/05/2022, 03:57 AM)Catullus Wrote: Please let  equal , where  is defined for  as the Bouncing Factorial of x, but instead of starting at one you start at two and you replace the multiplication in the definition with (y+2)-ation. Please let  equal . Please let  equal . I define my number as h^Guppyplex(Guppyplex)+1. Wow Catullus, your dedication to nice typesetting is impressive. In my experience many problems in physics and mathematics have answers because they mean something. Even more, the best problems are at the perimeter of our understanding. Asking questions is great, but some of your questions are far beyond the limit of our collective progress. Understanding the motivation behind a problem is usually the first step in solving it. Paul Erdős talked about "The Book", of the ultimate mathematics theorems. This is a good guide for what to work on. Sorry, I have no idea as to how to answer your question. Great response, Daniel. I'd say the same thing Catullus; and I mean it as a compliment; your questions are on or near the perimeter. And that's what's needed from a good mathematician. Catullus Fellow   Posts: 213 Threads: 47 Joined: Jun 2022 08/08/2022, 09:53 AM Oh, thank you. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ marcokrt Junior Fellow  Posts: 24 Threads: 3 Joined: Dec 2011 08/16/2022, 05:44 AM (This post was last modified: 08/16/2022, 05:52 AM by marcokrt.) (07/10/2022, 03:35 AM)Catullus Wrote: Can you please tell me some salad numbers that use hyper-operations? If you want, you may make up your own hyper-operational salad numbers and post them here. A salad number is a large number created with a mishmash of numbers or functions. Even if it is quite embarassing, here is my old (ugly) salad number: My first salad number Anyway, I have just discovered that somebody there has coined an extension of the above: Salad number inspired to my old salad number Let $$G(n)$$ be a generic reverse-concatenated sequence. If $$G(1) \notin \{2, 3, 7\}$$, then $$^{G(n)}G(n) \pmod {10^d}≡^{G({n+1})}G({n+1}) \pmod {10^d}$$, $$\forall n \in \mathbb{N}-\{0\}$$ ("La strana coda della serie n^n^...^n", p. 60). Daniel Fellow   Posts: 248 Threads: 80 Joined: Aug 2007 08/16/2022, 06:32 AM Let x >= 2, then determine the fixed point x such that, $$x=x \uparrow^x x$$ Daniel Catullus Fellow   Posts: 213 Threads: 47 Joined: Jun 2022 08/16/2022, 10:39 AM (08/16/2022, 06:32 AM)Daniel Wrote: Let x >= 2, then determine the fixed point x such that, $$x=x \uparrow^x x$$ Could you please explain what way is that related to hyper-operational salad numbers? Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ Daniel Fellow   Posts: 248 Threads: 80 Joined: Aug 2007 08/16/2022, 02:51 PM (08/16/2022, 10:39 AM)Catullus Wrote: (08/16/2022, 06:32 AM)Daniel Wrote: Let x >= 2, then determine the fixed point x such that, $$x=x \uparrow^x x$$ Could you please explain what way is that related to hyper-operational salad numbers? Sorry, if you supply be a salad metric I will try again. Daniel Catullus Fellow   Posts: 213 Threads: 47 Joined: Jun 2022 09/17/2022, 01:15 AM (This post was last modified: 09/17/2022, 03:38 AM by Catullus.) I changed the definition of my salad number and gave it a name. I might change the definition again, so you might want check in on the salad number sometimes. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ « Next Oldest | Next Newest »

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