aarex2

(***Original Text by Aarex Tiaokhiao***)

***The following is the original version of Aarex Tiaokhiao's Nested Cascading-E Notation with BEAF numbers. Here he includes many numbers.

Nested Cascading-E Notation with BEAF numbers

As a Cascading-E Notation, he reaches Great and Terrible Tethrathoth.

So here a extension of Cascading-E Notation:

A new rules is Uses a #^…^(#...#) = #^…(#^…^(#...)).

Nested Cascading-E Notation (E^…)

Other name “Cascading Hyper-E Notation”

E(a)b = a^b

# 1 = #

# n#m = # (# n#m-1)

# #¹m = # #m

# n #^lm = # n #^l-1 n...n #^l-1 n (m n's)

# n #^# m = # n #...# m (n #'s)

# n #^^@ m = n #^#...#^# (@ #^#'s)

# n #^...^@ m = # n #^...(#^...^@-1) m

# n @/# m = # n @-1 m

... = #^^# = #^# --> X

#^#*# --> X+1

#^#*## --> X+2

#^#*### --> X+3

...

#^#*#^# --> 2X

#^#*#^#*# --> 2X+1

#^#*#^#*## --> 2X+2

...

#^#*#^#*#^# --> 3X

...

#^#*#^#*#^#* ... *#^#*#^# w/n #^#s --> nX

I defined reverse-cascading symbol: */* --> *-*

Next we have #^## --> XX = X². So the #^## has the same power as planar arrays. Continuing on we have:

#^## --> X²

#^##*# --> X²+1

#^##*## --> X²+2

...

#^##*#^# --> X²+X

#^##*#^#*# --> X²+X+1

...

#^##*#^#*#^# --> X²+2X

#^##*#^#*#^#*#^# --> X²+3X

...

#^##*#^## --> 2X²

#^##*#^##*#^## --> 3X²+X

...

#^### --> X³

#^#### --> X⁴

...

#^^## = #^#^# --> X^X

Here we observe an interesting property of the comparison. The number of hashes in the exponent is also the number of dimensions of the comparable array. We can say if we have #^$(#) where $(#) is some expression composed of hyperions, replace all of these with X and we can say:

#^$(#) --> $(X)

Hence...

#^^### = #^#^#^# --> X^X^X

#^^#### = #^#^#^#^# --> X^X^X^X

#^^##### = #^#^#^#^#^# --> X^X^X^X^X

...

This notation reaches a #^^#^#, so here more:

#^^#^# = #^#...#^# (X ^'s) --> X^^X

#^(#^^(#^#/#)*#) --> X^(X^^(X-1)+1)

#^^(#^#*#) --> X^^(X+1)

#^^(#^#*##) --> X^^(X+2)

#^^(#^#*###) --> X^^(X+3)

#^^(#^#*#^#) --> X^^2X

#^^(#^#*#^#*#^#) --> X^^3X

#^^(#^##) --> X^^(X^2)

#^^(#^###) --> X^^(X^3)

#^^(#^#^#) --> X^^(X^X)

#^^(#^#^#^#) --> X^^(X^X^X)

#^^#^^#^# --> X^^X^^X

#^^#^^#^^#^# --> X^^X^^X^^X

How about pentation?

#^^^#^# --> X^^^X

Note: #^^^#^# = #^^#^^...#^^#^^#^# (X-1 ^^'s)

And hexation, heptation, ...?

#^^^^#^# --> X^^^^X, #^^^^#^# = #^^^#^^^...#^^^#^^^#^# (X-1 ^^^'s)

#^^^^^#^# --> X^^^^^X, #^^^^^#^# = #^^^^#^^^^...#^^^^#^^^^#^# (X-1 ^^^^'s)

In FGH, this growth rate is f_φ(_ω_,0)(n) > E(n)n#^...^#^#n /w n+1 ^'s.

As Extensible-E System, Nested Cascading-E Notation are fourth term.

As Hyper-E Notation is first (E#), Extended Hyper-E Notation (xE#) is second and Cascading-E Notation (E^) are third.

Formly New E#^...

Get ready to reaching #^...^#^#...

Tethrathoth Group (Expanded)

Now revisit of tethrathoth:

Tethrathoth = E100#^^#^#100

Did you know: #^^# in Extended Cascading-E Notation is equal to #^^#^#

Grand Tethrathoth = E100#^^#^#100#2

Grantethrathoth = E100#^^#^#100#100

Greatethrathoth = E100#^^#^#100#100#100

Gigantethrathoth = E100#^^#^#100##4

Gorgetethrathoth = E100#^^#^#100##5

Gultethrathoth = E100#^^#^#100##6

Gasptethrathoth = E100#^^#^#100##7

Ginortethrathoth = E100#^^#^#100##8

There a E100#^^#^#100##100 exist, so

gugold ~ tethrathoth =

Gugoltethrathoth = E100#^^#^#100##100

Greetagoltethrathoth = E100#^^#^#100##100#100

Gugolthratethrathoth = E100#^^#^#100##100##100

throogol ~ tethrathoth =

Throotethrathoth = E100#^^#^#100###100

Tristo-Throotethrathoth = E100#^^#^#100###100

teroogol ~ tethrathoth =

Terootethrathoth = E100#^^#^#100####100

Petootethrathoth = E100#^^#^#100#####100

Ectootethrathoth = E100#^^#^#100######100

There a Throotethrathoth series, so here a 100th member:

Godgahtethrathoth = E100#^^#^#100#^#100

Wait, there more!

Grandgahtethrathoth = E100#^^#^#100#^#100#100

Gugoldgahtethrathoth = E100#^^#^#100#^#100##100

Graatagoldgahtethrathoth = E100#^^#^#100#^#100##100#100

Gugolthragahtethrathoth = E100#^^#^#100#^#100##100##100

Teroogahtethrathoth = E100#^^#^#100#^#100###100

Gotrigahtethrathoth = E100#^^#^#100#^#100#^#100

Gotergahtethrathoth = E100#^^#^#100#^#100#^#100#^#100

We define a grid, throo, teroo, ...

Godgridgahtethrathoth = E100#^^#^#100#^#*#100

Godthroogahtethrathoth = E100#^^#^#100#^#*##100

Godteroogahtethrathoth = E100#^^#^#100#^#*###100

Deutero-godgahtethrathoth = E100#^^#^#100#^#*#^#100

Trito-godgahtethrathoth = E100#^^#^#100#^#*#^#*#^#100

Gridgahtethrathoth = E100#^^#^#100#^##100

Gridtrigahtethrathoth = E100#^^#^#100#^##100#^##100

Gridtergahtethrathoth = E100#^^#^#100#^##100#^##100#^#100

Gridgodgahtethrathoth = E100#^^#^#100#^##*#100

Gridgrandgahtethrathoth = E100#^^#^#100#^##*##100

Gridgugoldgahtethrathoth = E100#^^#^#100#^##*#^#100

Gridthroogahtethrathoth = E100#^^#^#100#^##*#^#*#^#100

Gridugahtethrathoth = E100#^^#^#100#^##*#^##100

Kubikahtethrathoth = E100#^^#^#100#^###100

Godgatethrathoth = E100#^^#^#100#^#^#100

Godtothrathoth = E100#^^#^#100#^#^#^#100

We define a tetrithrathoth:

Tetrithrathoth = E100#^^#^#100#^^#^#100

Teterthrathoth = E100#^^#^#100#^^#^#100#^^#^#100

Tetppegathrathoth = E100#^^#^#100#^^#^#100#^^#^#100#^^#^#100

We define a tetgoldthrathoth:

Tetgoldthrathoth = E100#^^#^#*#100

Tetgridthrathoth = E100#^^#^#*##100

Tethroothrathoth = E100#^^#^#*###100

Teteroothrathoth = E100#^^#^#*####100

Tetpetoothrathoth = E100#^^#^#*#####100

Tetectoothrathoth = E100#^^#^#*######100

Tetzettoothrathoth = E100#^^#^#*#######100

Tetyottothrathoth = E100#^^#^#*########100

We define a deutero-tethrathoth and trito-tethrathoth:

Deutero-tethrathoth = E100#^^#^#*#^^#^#100

Trito-tethrathoth = E100#^^#^#*#^^#^#*#^^#^#100

This is end of expanded tethrathoth group.

Tethrathor Group

Okay, we imagined hecato-tethrathoth.

This is Tethrator = E100#^(#^^(#^#/#)*#)100

At short one, @ is #^^(#^^#/#)*#, so tethrator = E100#^@100.

Wait there more!

Grand Tethrator = E100#^@100#2

Grantethrator = E100#^@100#100

Greatethrator = E100#^@100#100#100

Gugoltethrator = E100#^@100##100

Throothrator = E100#^@100###100

We define a tetrithrator = E100#^@100#^@100, teterthrator = E100#^@*#4 and tetppegathrator = E100#^@*#5.

We define a tetgoldthrator = E100#^@*#100 and tetgridthrator = E100#^@*##100.

We define deutero-tetgoldthrator = E100#^@*#^@100 and trito-todathoth = E100#^@*#^@*#^@100.

Tetgridthrathor Group

Here a tetgridthrathor. Its a E100#^(#^^###...###*##)100

There more:

Grand Tetgridthrathor = E100#^(#^^(#^^#/#)*##)100#2

Grantetgridthrathor = E100#^(#^^(#^^#/#)*##)100#100

Tetrigridthrathor = E100#^(#^^(#^^#/#)*##)*#3

Tetergridthrator = E100#^(#^^(#^^#/#)*##)*#4

Tetppegagridthrator = E100#^(#^^(#^^#/#)*##)*#5

Deutero-tetgridthrathor = E100#^(#^^(#^^#/#)*##)*#^(#^^(#^^#/#)*##)100

Trito-tetgridthrathor = E100#^(#^^(#^^#/#)*##)*#^(#^^(#^^#/#)*##)*#^(#^^(#^^#/#)*##)100

Tethroothrathor Group

A tethroothrathor is equal to E100#^(#^^(#^#/#)*###)100

Teteroothrathor is equal to E100#^(#^^(#^#/#)*####)100

Tetpetoothrathor is equal to E100#^(#^^(#^#/#)*#####)100

Tetectoothrathor is equal to E100#^(#^^(#^#/#)*######)100

Tetzettoothrathor is equal to E100#^(#^^(#^#/#)*#######)100

Tetyottoothrathor is equal to E100#^(#^^(#^#/#)*########)100

Using a terrible tethrathoth series:

Tethrathoth = E100#^^#^#100

Terrible Tethrathoth = E100(#^^#^#)^^#^#100

Terrible Terrible Tethrathoth = E100((#^^#^#)^^#^#)^^#^#100

...

Tethrathoth ba'al = E100(...#^^#^#...)^^#^#100 /w 100 ^^#^#'s

...

Great and Terrible Tethrathoth = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s

...

Great and Terrible Tethrathoth ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s

...

Great and Terrible Tethrathoth super ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s

...

Great and Terrible Tethrathoth duper ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s

...

Great and Terrible Tethrathoth tripler ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s

...

Great and Terrible Tethrathoth teripler ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s

Tethrathordeus = E100#^(#^^(#^#/#))100

Tethrathortruce = E100#^(#^^(#^#/#)*#^^(#^#/#)*#^^(#^#/#))100

Tethrathorquad = E100#^(#^^(#^#/#)*#^^(#^#/#)*#^^(#^#/#)*#^^(#^#/#))100

Todatbiath Group

Todatbiath = E100#^#^(#^^###...###*#)100 \w 98 #'s

Trandatbiath = E100#^#^(#^^###...###*##)100 \w 98 #'s

Tugolthratbiath = E100#^#^(#^^###...###*#^#)100 \w 98 #'s

Tugoltriatbiath = E100#^#^(#^^###...###*#^##)100 \w 98 #'s

Throogolthratbiath = E100#^#^(#^^###...###*#^###...###)100 \w 98 #'s

Tris-throogolthratbiath = E100#^#^(#^^###...###*#^###...###*#^###...###)100 \w 98 #'s

Tessi-throogolthratbiath = E100#^#^(#^^###...###*#^###...###*#^###...###*#^###...###)100 \w 98 #'s

Todatriath Group and Beyond

Todatriath = E100#^#^#^(#^^###...###*#)100 \w 97 #'s

Trandatriath = E100#^#^#^(#^^###...###*##)100 \w 97 #'s

Tugolthratriath = E100#^#^#^(#^^###...###*#^#)100 \w 97 #'s

Throogolthratriath = E100#^#^#^(#^^###...###*#^###...###)100 \w 97 #'s

Tristo-throogolthratriath = E100#^#^#^(#^^###...###*#^###...###*#^###...###)100 \w 97 #'s

Teristo-throogolthratriath = E100#^#^#^(#^^###...###*#^###...###*#^###...###*#^###...###)100 \w 97 #'s

Todatetrath = E100#^#^#^#^(#^^###...###*#)100 \w 96 #'s

Trandatetrath = E100#^#^#^#^(#^^###...###*##)100 \w 96 #'s

Tulthratetrath = E100#^#^#^#^(#^^###...###*#^#)100 \w 96 #'s

Throogolthratetrath = E100#^#^#^#^(#^^###...###*#^###...###)100 \w 96 #'s

Tristo-throogolthratetrath = E100#^#^#^#^(#^^###...###*#^###...###*#^###...###)100 \w 96 #'s

Teristo-throogolthratetrath = E100#^#^#^#^(#^^###...###*#^###...###*#^###...###*#^###...###)100 \w 96 #'s

Beyond Tessi-throogolthratetrath

Gridthomegath = E100#^#...#^#^##100 \w 100 #'s

Throomegath = E100#^#...#^#^###100 \w 100 #'s

Tethomkillath = E100#^#...#^#^#^#100 \w 101 #'s

Gridthomkillath = E100#^#...#^#^#^##100 \w 101 #'s

Throomkillath = E100#^#...#^#^#^###100 \w 101 #'s

Tethomath = E100#^#...#^#^#^#^#100 \w 102 #'s

Tethomgigath = E100#^#...#^#^#^#^#^#100 \w 103 #'s

Tethomegator = E100#^#...#^#...#^#100 \w 200 #'s

Tethomkillator = E100#^#...#^#100 \w 201 #'s

Tethomator = E100#^#...#^#100 \w 202 #'s

Tethomegatohol = E100#^#...#^#...#^#100 \w 300 #'s

Tethomegatethol = E100#^#...#^#...#^#100 \w 400 #'s

Tethomegathousath = E100#^#...#^#...#^#100 \w 10000 #'s

Tethomegongath = E100#^#...#^#...#^#100 \w 1000000 #'s

Tethomegingath = E100#^#...#^#...#^#100 \w 100000000 #'s

Tethomegoath = E100#^#...#^#...#^#100 \w 100^100 #'s

Tethomegiath = E100#^#...#^#...#^#100 \w 100^100^100 #'s

Tethomegaath = E100#^#...#^#...#^#100 \w 100^100^100^100 #'s

Tethrathothatria Group

Ok the 1st group is godgahlah group and 2nd group is tethrathoth group.

This a 3rd group:

Tethrathothatria = E100#^^#^^#^#100

Tethomegoathatria = E100#^^#^^#^#^#100

Tethomegiathatria = E100#^^#^^#^#^#^#100

Tethomegaathatria = E100#^^#^^#^#^#^#^#100

Tethrathothatetra Group to Tethrathothaocta Group

For 4th group, we put -atetra at a end:

Tethrathothatetra = E100#^^#^^#^^#^#100

Tethomegoathatetra = E100#^^#^^#^^#^#^#100

Tethomegiathatetra = E100#^^#^^#^^#^#^#^#100

Tethomegaathatetra = E100#^^#^^#^^#^#^#^#^#100

For 5th group, we put -apenta at a end:

Tethrathothapenta = E100#^^#^^#^^#^^#^#100

Tethomegoathapenta = E100#^^#^^#^^#^^#^#^#100

Tethomegiathapenta = E100#^^#^^#^^#^^#^#^#^#100

Tethomegaathapenta = E100#^^#^^#^^#^^#^#^#^#^#100

For 6th group, we put -ahexa at a end:

Tethrathothahexa = E100#^^#^^#^^#^^#^^#^#100

Tethomegoathahexa = E100#^^#^^#^^#^^#^^#^#^#100

Tethomegiathahexa = E100#^^#^^#^^#^^#^^#^#^#^#100

Tethomegaathahexa = E100#^^#^^#^^#^^#^^#^#^#^#^#100

For 7th group, we put -ahepta at a end:

Tethrathothahepta = E100#^^#^^#^^#^^#^^#^^#^#100

Tethomegoathahepta = E100#^^#^^#^^#^^#^^#^^#^#^#100

Tethomegiathahepta = E100#^^#^^#^^#^^#^^#^^#^#^#^#100

Tethomegaathahepta = E100#^^#^^#^^#^^#^^#^^#^#^#^#^#100

And for 8th group, we put -aocta at a end:

Tethrathothaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#100

Tethomegoathaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#^#100

Tethomegiathaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#^#^#100

Tethomegaathaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#^#^#^#100

Penthrathoth Group

Ok, let godgahlah for stage 1, tethrathoth for stage 2, tethrathothatria for stage 3, tethrathothatetra for stage 4,...

Let number is stage 100, penthrathoth which equal to E100#^^^#^#100 and this number is comparable from Bower's kungulus ({10,100,3} & 10).

So grand penthrathoth is equal to E100#^^^#^#100#2 (This number is comparable from Bower's kungulusplex ({10,kungulus,3} & 10)), grand grand penthrathoth is equal to E100#^^^#^#100#3 and granpenthrathoth is equal to E100#^^^#^#100#100,

Ok, here more.

greapenthrathoth = E100#^^^#^#100#100#100

giganpenthrathoth = E100#^^^#^#100#100#100#100

gorgepenthrathoth = E100#^^^#^#100##5

...

gugolpenthrathoth = E100#^^^#^#100##100

graatagolpenthrathoth = E100#^^^#^#100##100#100

greegolpenthrathoth = E100#^^^#^#100##100#100#100

...

gugoltriapenthoth = E100#^^^#^#100##100##100

gugoltelsapenthoth = E100#^^^#^#100##100##100##100

gugolpentapenthoth = E100#^^^#^#100##100##100##100##100

throopenthrathoth = E100#^^^#^#100###100

teroopenthrathoth = E100#^^^#^#100####100

godgahpenthrathoth = E100#^^^#^#100#^#100

tethrapenthrathoth = E100#^^^#^#100#^^#^#100

penthratrithoth = E100#^^^#^#100#^^^#^#100

deutero-penthrathoth = E100#^^^#^#*#^^^#^#100

trito-penthrathoth = E100#^^^#^#*#^^^#^#*#^^^#^#100

penthrathothatria = E100#^^^#^^^#^#100.

penthrathothatetra = E100#^^^#^^^#^^^#^#100

penthrathothapenta = E100#^^^#^^^#^^^#^^^#^#100

Hexathrathoth Group

A hexathrathoth = E100#^^^^#^#100. This number is comparable from Bower's quadrunculus ({10,100,4} & 10).

hexathrathothatria = E100#^^^^#^^^^#^#100. This number is comparable from my quadrunculusplex.

Quadrunculusplex = {10,quadrunculus,4} & 10 and has geegolplex entries.

hexathrathothatetra = E100#^^^^#^^^^#^^^^#^#100

A hepthrathoth = E100#^^^^^#^#100. This number is comparable from my quintunculus.

Quintunculus = {10,100,5} & 10 and has gigol entries.

hepthrathothatria = E100#^^^^^#^^^^^#^#100

hepthrathothatetra = E100#^^^^^#^^^^^#^^^^^#^#100

A octhrathoth = E100#^^^^^^#^#100. This number is comparable from my sextunculus.

Sextunculus = {10,100,6} & 10 and has goggol entries.

octhrathothatria = E100#^^^^^^#^^^^^^#^#100

octhrathothatetra = E100#^^^^^^#^^^^^^#^^^^^^#^#100

A nonthrathoth = E100#^^^^^^^#^#100. This number is comparable from my septunculus.

Septunculus = {10,100,7} & 10 and has gagol entries.

nonthrathothatria = E100#^^^^^^^#^^^^^^^#^#100

nonthrathothatetra = E100#^^^^^^^#^^^^^^^#^^^^^^^#^#100

...

Keep a up!

Omegathrathoth Group

Now beyond!

Godgathor = E100#^#^#100

Tethrathoth = E100#^^#^#100

Penthrathoth = E100#^^^#^#100

Hexathrathoth = E100#^^^^#^#100

Hepthrathoth = E100#^^^^^#^#100

Octhrathoth = E100#^^^^^^^#^#100

Nonthrathoth = E100#^^^^^^^^#^#100

Decthrathoth = E100#^^^^^^^^^#^#100

Undecthrathoth = E100#^^^^^^^^^^#^#100

...

Omegathrathoth = E100#^^...^^#^#100 \w 100 #'s. This number is comparable from Bower's humongulus ({10,10,100} & 10).

Wait, there a bigger number, Omegathrathothigong = E100000#^^...^^#^#100000 \w 100000 #'s

Wait, there MORE bigger, Grand Omegathrathoth = E100#^^...^^#^#100 \w omegathrathoth #'s. This number is comparable from my humongulusplex.

Humongulusplex = {10,10,humongulus} & 10 and has {10,10,humongulus} entries.

Wait, there and a biggest Nested Cascading-E notation number, Grand Omegathrathothigong = E100000#^^...^^#^#100000 \w Omegathrathothigong #'s.

What? This is limited of NCEN! We continue this, using xE^...?!

Granomegathrathoth = E100#$#^#100#100

Gugolomegathrathoth = E100#$#^#100##100

Throomegathrathoth = E100#$#^#100###100

Godgahomegathrathoth = E100#$#^#100#^#100

Tethraomegathrathoth = E100#$#^#100#^^#^#100

Penthraomegathrathoth = E100#$#^#100#^^^#^#100

Omegathraomegathrathoth = E100#$#^#100#$#^#100

Omegathraomegathraomegathrathoth = E100#$#^#100#$#^#100#$#^#100

Until ...-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathrathoth.

Let called Extended Nested Cascading-E Notation. This is still not far enough.

As NEW Expand-E Notation (E^<#>), ^ change to <#>, ^^...^^ (N ^'s) change to <##...##> (N #'s), and ^^...^^ (X ^'s) or $, using Extended Nested Cascading-E Notation change to <#^#>.

As Extensible-E System, Expand-E Notation are fifth term.

(Modified Version)

***Later Aarex modified the article and deleted most of his googolisms.

Nested Cascading-E Notation with BEAF numbers

As a Cascading-E Notation, he reaches a Great and Terrible Tethrathoth.

So here a extension of Cascading-E Notation:

A new rules is Uses a @^...^# = #^...(#^...^(#...)).

Nested Cascading-E Notation (E^...)

Other name "Cascading Hyper-E Notation"

E(a)b = a^b

# 1 = #

# n#m = # (# n#m-1)

# #¹m = # #m

# n #^lm = # n #^l-1 n...n #^l-1 n (m n's)

# n @^# m = # n @*@...@*@ m (n @'s)

# n @^^...^^# m = n @^^...^@...@^^...^@ n (m-1 ^^...^'s)

# n @^...^(@ #) m = # n (@^...^@)^^# m

#^# --> X

#^#*# --> X+1

#^#*## --> X+2

#^#*### --> X+3

...

#^#*#^# --> 2X

#^#*#^#*# --> 2X+1

#^#*#^#*## --> 2X+2

...

#^#*#^#*#^# --> 3X

...

#^#*#^#*#^#* ... *#^#*#^# w/n #^#s --> nX

Next we have #^## --> XX = X². So the #^## has the same power as planar arrays. Continuing on we have:

#^## --> X²

#^##*# --> X²+1

#^##*## --> X²+2

...

#^##*#^# --> X²+X

#^##*#^#*# --> X²+X+1

...

#^##*#^#*#^# --> X²+2X

#^##*#^#*#^#*#^# --> X²+3X

...

#^##*#^## --> 2X²

#^##*#^##*#^## --> 3X²+X

...

#^### --> X³

#^#### --> X⁴

...

#^^## = #^#^# --> X^X

#^$(#) --> $(X)

Hence...

#^^### = #^#^#^# --> X^X^X

#^^#### = #^#^#^#^# --> X^X^X^X

#^^##### = #^#^#^#^#^# --> X^X^X^X^X

...

This notation reaches a #^^#, so here more:

#^^#^# --> X^^X

(#^^#)^# --> (X^^X)^X

(#^^#)^(#^^#) --> (X^^X)^(X^^X)

(#^^#)^(#^^#)^(#^^#) --> (X^^X)^(X^^X)^(X^^X)

#^^## --> ~ X^^2X

#^^### --> ~ X^^3X

#^^#^# --> ~ X^^(X^2)

#^^#^## --> ~ X^^(X^3)

#^^#^#^# --> ~ X^^(X^X)

#^^#^#^#^# --> ~ X^^(X^X^X)

#^^#^^# --> ~ X^^X^^X

#^^#^^#^^# --> ~ X^^X^^X^^X

How about pentation?

#^^^# --> ~ X^^^X

#^^^## --> ~ X^^^2X

#^^^#^# --> ~ X^^^X^2

#^^^#^^# --> ~ X^^^X^^X

#^^^#^^^# --> ~ X^^^X^^^X

And hexation, heptation, ...?

#^^^^# --> ~ X^^^^X

#^^^^^# --> ~ X^^^^^X

In FGH, this growth rate is f_φ(_ω_,0)(n) > E(n)n#^...^#n /w n+1 ^'s.

As Extensible-E System, Nested Cascading-E Notation are fourth term.

As Hyper-E Notation is first (E#), Extended Hyper-E Notation (xE#) is second and Cascading-E Notation (E^) are third.

Formly New E#^...

Get ready to reaching #^...^#...

Tethrathoth Group (Expanded)

Now revisit of tethrathoth:

Tethrathoth = E100#^^#100

Did you know: #^^# in Extended Cascading-E Notation is same!

Grand Tethrathoth = E100#^^#100#2

Grantethrathoth = E100#^^#100#100

Greatethrathoth = E100#^^#100#100#100

Gigantethrathoth = E100#^^#100##4

Gorgetethrathoth = E100#^^#100##5

Gultethrathoth = E100#^^#100##6

Gasptethrathoth = E100#^^#100##7

Ginortethrathoth = E100#^^#100##8

There a E100#^^#100##100 exist, so

gugold ~ tethrathoth =

Gugoltethrathoth = E100#^^#100##100

throogol ~ tethrathoth =

Throotethrathoth = E100#^^#100###100

Tristo-Throotethrathoth = E100#^^#100###100

teroogol ~ tethrathoth =

Terootethrathoth = E100#^^#100####100

Petootethrathoth = E100#^^#100#####100

Ectootethrathoth = E100#^^#100######100

We defined tetrithrathoth = E100#^^#100#^^#100

teterthrathoth = E100#^^#100#^^#100#^^#100

Deutero-tethrathoth = E100#^^#*#^^#100

Trito-tethrathoth = E100#^^#*#^^#*#^^#100

This is end of expanded tethrathoth group.

Terriblethrathoth Group

To define Terriblethrathoth, we need #^^## seperator. E100#^^##100 = E100(#^^#)^^#100 = E100(#^^#)^(#^^#)...(#^^#)^(#^^#)100 (with 100 #^^#'s) = ...

So Terriblethrathoth (old is Terrible Tethrathoth) is equal to E100#^^##100.

Then Thrthrathoth (old is Terrible Terrible Tethrathoth) = E100#^^###100

Terthrathoth (old is Terrible Terrible Tethrathoth) = E100#^^####100

Pethrathoth (old is Terrible Terrible Terrible Tethrathoth) = E100#^^#####100

Ecthrathoth = E100#^^######100

Zetthrathoth = E100#^^#^#7

And Yotthrathoth = E100#^^#^#8.

Tethrathoth ba'al is E100#^^#^#100 and Great and Terrible Tethrathoth is E100#^^#^#100#2.

Wait, there more!

Great and Terrible Tethrathoth ba'al is E100#^^#^#100#2

Great and Terrible Tethrathoth super ba'al is E100#^^#^#100#3

Great and Terrible Tethrathoth duper ba'al is E100#^^#^#100#4

Great and Terrible Tethrathoth truper ba'al is E100#^^#^#100#5

...

Tethrathor Group

Tethrathor = E100#^^#^^#100

Terriblethrathor = E100#^^#^^##100

Thrthrathor = E100#^^#^^###100

Review

We need new numbers!

Let called Extended Nested Cascading-E Notation. This is still not far enough.

As NEW Expand-E Notation (E^<#>), ^ change to <#>, ^^...^^ (N ^'s) change to <##...##> (N #'s), and ^^...^^ (X ^'s) or $, using Extended Nested Cascading-E Notation change to <#^#>.

As Extensible-E System, Expand-E Notation are fifth term.

Expand-E Notation

Okay, as extension of Nested Cascading-E Notation, here it a Expand-E Notaion!

Expand-E Notation (E^<#>)

Here a new rules:

@ a #^...<@ #>@ b = @ a #^...^<@>@ b

#^...^<>@ = #^...^@

@ a #^...^@ b = @ a #*@ b, if 0 ^'s

#<#^#>#^# = X^...^X (X ^'s)

#<#^#>#<#^#>#^# = X^...^X^...^X (X ^'s between X)

#<#^#>#<#^#>#<#^#>#^# = X^...^X^...^X^...^X (X ^'s between X)

#<#^#*#>#^# = X^...^X (X+1 ^'s)

#<#^#*#^#>#^# = X^...^X (X*2 ^'s)

#<#^#*#^#*#^#>#^# = X^...^X (X*3 ^'s)

New Numbers

Omegathrathoth Group

Omegathrathoth = E100#<#^#>#^#100

Grand Omegathrathoth = E100#<#^#>#^#100#2

Granomegathrathoth = E100#<#^#>#^#100#100

Greaomegathrathoth = E100#<#^#>#^#100#100#100

Gugolomegathrathoth = E100#<#^#>#^#100##100

Throomegathrathoth = E100#<#^#>#^#100###100

Godgahomegathrathoth = E100#<#^#>#^#100#^#100

Array-E Notation

This is extension of Expand-E Notation. Array-E Notation (E#<...>)

I added 2 rules:

@ a #<,...,#...## @>#...## b = @ a #<,... #<,...,#...## @>#...#,#...# @>#^# b

@ a #<...,#>#...# b = @ a #<...>#...# b