Copyright 2016-2020 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)

5 *u* 2 = 5^5^3125

Clearly the results become unwritable in short order.

(the final exponent in 6 u 3 would only be obtained by raising 6 to
an exponent with over 36,000 digits).

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).

The principle that each rising exponent is the value of the entire tower below it yields the formula

n u b = n^^2^b

so those inclined to tetration(^^) with its constant exponents are rapidly left behind by the exponentially increasing exponents of ultrex; an ultrex bound of 2000 is sufficient to produce higher value than tetration to a centilliard(10^603); one of 19929 surpasses tetration to a kilillion(1000000^1000 or 10^6000);one of 199313 surpasses tetration to a deckilillion(1000000^10000 or 10^60000).

In Conway arrow terms n u b can be expressed as n→(2→b)→2 .

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**)^{
(n u b)}
*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.

A "captured" subscripted left bracket,isolated by spaces,represents the generation of the number indicated by the subscript...

**n** ( _{[n u b]}( ) *u*) **b**

for example.

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

In the meantime,take a look at the new Hyper-Operating
Ultrex Function which builds on basic ultrexing,and the
Simple Hyper-Operating Transformer Function and
Ladder Hyper-Operating Transformer Function
which go further.
*u*
*u*
*u*
__u____(
__ .
__
[
__ .
__
{__

This revision October 29,2020 (Conway comparison added).