## The Ultrex Function

#### Part of Counting Really,Really,Really High

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)
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 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 (n u b) (n u b) 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.

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).