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 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 \(\}[n]={}^n\ \).
(08/07/2022, 06:02 AM)Daniel Wrote: [ -> ] (08/05/2022, 03:57 AM)Catullus Wrote: [ -> ]Please let ![[Image: png.image?\dpi%7B110%7D%20f(x)]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20f(x))
equal ![[Image: png.image?\dpi%7B110%7D%20ceil(\tau\uparrow\...ambda%5E%7B+++]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20ceil(\tau\uparrow\uparrow(\pi\uparrow\uparrow(x+1)))\Lambda%5E%7B+++)
, where ![[Image: png.image?\dpi%7B110%7D%20x\Lambda%5E%7B\underbr...cdots++%7D_y]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20x\Lambda%5E%7B\underbrace%7B++\cdots++%7D_y)
is defined for ![[Image: png.image?\dpi%7B110%7D%20y\geq1]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20y\geq1)
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 ![[Image: png.image?\dpi%7B110%7D%20g(x)]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20g(x))
equal ![[Image: png.image?\dpi%7B110%7D%20f%5Ex(x)\Lambda%5E%7B\un...+%7D_%7Bf%5Ex(x)]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20f%5Ex(x)\Lambda%5E%7B\underbrace%7B++\cdots++%7D_%7Bf%5Ex(x))
.
Please let ![[Image: png.image?\dpi%7B110%7D%20h(x)]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20h(x))
equal ![[Image: png.image?\dpi%7B110%7D%20g%5Ex(x)\uparrow%5E%7Bg%5E...)%7D%7D%7Dg%5Ex(x)]](https://latex.codecogs.com/png.image?\dpi%7B110%7D%20g%5Ex(x)\uparrow%5E%7Bg%5E%7Bx+2%7D(x+2)\Lambda%5E%7B\underbrace%7B++\cdots++%7D_%7Bg%5E%7Bx+4%7D(x+4)%7D%7D%7Dg%5Ex(x))
.
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.
(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 x >= 2, then determine the fixed point x such that,
\(x=x \uparrow^x x\)
(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?
(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.
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.