Copyright 2016 by Louis Epstein,All Rights Reserved

The Ultrex Function,or

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** _{99}*u* **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
**n***u***b** *u*'s between **n** and **b**.

A second underlined *u* means
(**n**u**b**)^{
(nub)}
*u*'s between **n** and **b**.

A third underlined *u* means
(**n**u**b**)^{
(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
(**n**u**b**)^{
(nub)} layers of nested (on the bound)
**n**u**b**)'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____(
__ .
__
[
__ .
__
{__