The Ultrex Function

Part of Counting Really,Really,Really High


Copyright 2016 by Louis Epstein,All Rights Reserved
The Ultrex Function,or u,raises a base n to a series of exponents numbering as defined by the bound b.
The first exponent is always n itself,so n u 1 equals nn for any value of n.Each succeeding exponent is the solution of the entire power tower below it:

2 u 2 = 2^2^(2^2) = 2^2^4 = 2^16 = 65536
3 u 2 = 3^3^27 = 3^7625597484987
2 u 3 = 2^2^(2^2)^(2^2^(2^2)) = 2^2^4^65536
4 u 2 = 4^4^256 = 4^13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
3 u 3 = 3^3^27^(3^7625597484987)

Clearly the results become unwritable in short order.
10 u 2 is about the 9,999,999,920th power of the number of subatomic particles in the material universe.(2 u 10 is much greater,as growth in the bound far more heavily influences the result than growth in the base).

Still,there needs to be massive expansion of this notation in order to reach really big numbers.

Repetitions of u:
X (n u's) Y = X (n-1 u's) (X u Y).
Thus
2 uu 2 = 2 u 65536.

Simple multiplication can be done with a subscript:
n 99u b means 99 u's.

However,we're out to smash the heavens of large-numberdom, so we need special notations.
n u b means that there are nub u's between n and b.

A second underlined u means (nub) (nub) u's between n and b.

A third underlined u means (nub) (nub) ((nub) (nub)) u's between n and b...and so on in ultrex fashion.

n (u) b
means there are n (n u b u's) b u's between n and b.

A second,third etc. set of parentheses likewise ultrex the number of u's in the specification of the number thereof.

n [u] b
means there are n (n u (n u ... (n u b layers of nesting) b ) u's) b u's between n and b.

A second,third etc. set of square brackets likewise ultrex the number of layers of nesting in the specification of the number of u's.

Substituting braces {} for the square brackets replaces every u in the nested specification with (nub) (nub) layers of nested (on the bound) nub)'s.

In every succeeding generation of bracketing,the number of substitutions deepening the inner specification increases in ultrex fashion.After {u} the generations are
(u)
(u)
(u)
[u]
[u]
[u]
{u}
{u}
{u}
(u)

Followed by this whole succession with underlined u.

More to come on using ultrex functions to specify the numbers of generations of bracketing.

u u u ( . [ . {