User blog:Allam948736/Caret-Star Notation

I invented another new notation that generalizes Steinhaus-Moser notation and achieves BEAF-level growth rates. The first delimiter, a(*^b)c, is defined like so:

a(*^2)b = a^b a(*^3)b = \(a^{a^b}\) a(*^n)b = \(a^{a^{...^{a^b}}}\) w/ n-1 copies of a a(*^*)b = a(*^b)b

a(*^^2)b = a^a repeated b times (equal to a in b triangles in Steinhaus-Moser notation). a(*^^3)b = \(a \uparrow\uparrow 3\) repeated b times (equal to a in b triangles in a version of Steinhaus-Moser notation where n[3] = n^(n^n) instead of just n^n) a(*^^n)b = \(a \uparrow\uparrow n\) repeated b times a(*^^*)b = \(a \uparrow\uparrow a\) repeated b times

a(*^^^2)b = a(*^^2)a repeated b times (this is equivalent to a in b squares in Steinhaus-Moser notation)

Mega can be expressed exactly as 2(*^^^2)2 or 2(*^^^^2)1.

I defined a series of numbers analogous to the googol, starting with bidugol, which is equal to 10(*^^2)100 (approximately E100,000,000,011#99). I then named the tridugol, which is equal to 10(*^^3)100 (approximately E10,000,000,010#199), the quadugol (10(*^^4)100), and the quindugol, which is equal to 10(*^^5)100.

The hypermega is equal to 2(*^^^3)2. I actually came up with the name back in July 2015 (well before I created my site, let alone joined this wiki). This number is approximately \(10 \uparrow\uparrow (2^{2^{66} + 1} + 3)\) (or more precisely, about E(2.9374371566719126*10^22212093154093428548)#(2^(2^66 + 1)), and its last 10 digits are ...6,964,233,216.

Next, I defined a(*^[0]^b)c to be equal to a(*^^^...^^^b)a w/ c ^s, which diagonalizes over everything I covered up to this point. After this, the notation gets a bit complicated, so I linked my site's article for more. If my analysis is correct, the highest delimiter I have currently defined, *^[*, 3]^*, has a growth rate roughly equivalent to \(f_{\omega^{\omega^4}}(n)\) in the fast-growing hierarchy, on par with Bowers' 4-dimensional arrays.

Source:

https://sites.google.com/site/allamsnumbers/home/hierarchies-and-advanced-notations/caret-star-notation