Some months ago, or about 1 year ago? I calculated Moser's number, using up-arrow notation. Later I posted movie about Moser, to Nico-nico-douga:
However, those movies are in Japanese, so I want to write down the process of calculation, in English.
It needs some knowledges of approximation between up-arrow notation and Steinhaus-Moser notation.
Approximation section 1
we need some preparations to calculate moser.
First, let's consider about this.
- when a and n is not small.
This can be calculated as follows:
Here is so big, but
If b=a, this can be described as follows, using tetration operator,
using that step,
Because of this approximation, the next can be also approximated as follows,
so, using pentation operator,
Similarly and generally,
And of course,
Approximation section 2
For example, if n is not small,
The differences of exponential heights between tetration of 10, are only 1 or 2.
So if N>>0, for any a,
Nextly, for example,
- when a' is a, or a number near a.
So, if N>>0,
Those dispositions are necessary to calculate polygon notation.
Now let's consider about n in p-gon, as n[p]. n is not small (at least 3 or more).
- if n is not small
Here we are passing the first steps of approximation.
So when it reach n in square,
- and if N>>0,
if n' =: n+1,
- (using second steps of approximation)
...Finally, in general,
Calculation of 2[p]
2 is small, and so week, among up-arrow notation, so it's hard to apply . However, 2[p]=(2[p-1])[p-1], and if p is not small, 2[p-1] is large number. So (2[p-1])[p-1] can be applied approximation of n[p].
- = Mega
Another approach to 2[p]
now  is exponentiation, and  is approximately tetration, similarly  is pentation,  is hexation, and [p] is, approximately .
So concidering pattern of previous section,
And Moser's number is,
Can we approximate this more simply? Yes, there is two way to describe this structure.
can be written down as follows,
It is similar to 2[p]
Compareing these and Moser, in fact,
After all, However, Mega is already quite large number.
This might be the most simple approximation, using up-arrow notation...