saw this in another post and though I would comment since I'm feeling like rambling for a while
Pikers all. I got interested in big numbers once and started off with what I found out later was called the tower function by some;
a0,i = 1; i>0
aj,i= \(\displaystyle i^{a_{j-1,i}};\, i,\, j\, >\, 0\),
and finally
tower(i) = ai,i; i>0
For example
tower(1) = 1
tower(2) = 22 = 4
tower(3) = \(\displaystyle 3^{3^3}\) = 327 = 7 625 597 484 987
tower(4) = \(\displaystyle 4^{4^{4^4}}\) = \(\displaystyle 4^{4^{256}}\) ~ \(\displaystyle 4^{1.341 * 10^{154}}\) ~ \(\displaystyle 10^{1.215 * 10^{153}}\)
Although tower(4) is a number larger than the number of atoms in the know universe, the tower function is also small potatoes when it comes to big numbers.
My stab at some really big numbers is the cap t [T] function which may (or may not) be related to the Knuth's up-arrow numbers, i.e.
https://en.wikipedia.org/wiki/Knuth's_up-arrow_notation
and thus to other notations such as the Ackermann function:
First some background,
[an]= {\(\displaystyle (a_1, a_2, a_3, ..., a_n), a_i\, \epsilon\, \mathit{Z^+}\)}
[an,b]= {\(\displaystyle (a_1, a_2, a_3, ..., a_n, b), a_i,\, b\, \epsilon\, \mathit{Z^+}\)} = [an+1] where an+1 = b
t[a1] = tower(a1)
t[an, 1] = t[an], n > 0
t[a1, b] = t[t[a1,b-1], b-1], b>1
t[an, b] = t[an-1, t[an-1, b-1], b-1], b>1, n>1
Finally, let
[2n] = {(2, 2, 2, ..., 2)}, that is n 2's,
and
Tn = t[2n]
Examples
T1 = t[2] = tower(2) = 22 = 4
T2 = t[(2,2)] = t[t(2,1), 1] = t[t[2],1]=t[t[2]]=t[4]=tower(4)
T3 = t[(2,2,2)] = t[2,t[(2,2,1)],1] = = t[2,t[(2,2)]] = t[2,t[4]]
which I think we can agree is pretty darn big and we can stop there (although I did carry it one step further just to be ridiculous).
Pikers all. I got interested in big numbers once and started off with what I found out later was called the tower function by some;
a0,i = 1; i>0
aj,i= \(\displaystyle i^{a_{j-1,i}};\, i,\, j\, >\, 0\),
and finally
tower(i) = ai,i; i>0
For example
tower(1) = 1
tower(2) = 22 = 4
tower(3) = \(\displaystyle 3^{3^3}\) = 327 = 7 625 597 484 987
tower(4) = \(\displaystyle 4^{4^{4^4}}\) = \(\displaystyle 4^{4^{256}}\) ~ \(\displaystyle 4^{1.341 * 10^{154}}\) ~ \(\displaystyle 10^{1.215 * 10^{153}}\)
Although tower(4) is a number larger than the number of atoms in the know universe, the tower function is also small potatoes when it comes to big numbers.
My stab at some really big numbers is the cap t [T] function which may (or may not) be related to the Knuth's up-arrow numbers, i.e.
https://en.wikipedia.org/wiki/Knuth's_up-arrow_notation
and thus to other notations such as the Ackermann function:
First some background,
[an]= {\(\displaystyle (a_1, a_2, a_3, ..., a_n), a_i\, \epsilon\, \mathit{Z^+}\)}
[an,b]= {\(\displaystyle (a_1, a_2, a_3, ..., a_n, b), a_i,\, b\, \epsilon\, \mathit{Z^+}\)} = [an+1] where an+1 = b
t[a1] = tower(a1)
t[an, 1] = t[an], n > 0
t[a1, b] = t[t[a1,b-1], b-1], b>1
t[an, b] = t[an-1, t[an-1, b-1], b-1], b>1, n>1
Finally, let
[2n] = {(2, 2, 2, ..., 2)}, that is n 2's,
and
Tn = t[2n]
Examples
T1 = t[2] = tower(2) = 22 = 4
T2 = t[(2,2)] = t[t(2,1), 1] = t[t[2],1]=t[t[2]]=t[4]=tower(4)
T3 = t[(2,2,2)] = t[2,t[(2,2,1)],1] = = t[2,t[(2,2)]] = t[2,t[4]]
which I think we can agree is pretty darn big and we can stop there (although I did carry it one step further just to be ridiculous).