excelguru wrote:

ev3coolexit987654 wrote:

excelguru wrote:

TO: ev3coolexit987654

Okay, I am going to combine all of my functions:

Let N1 = 100{↑}_(100)100{↑}_(100)100 in my first extended up-arrow notation.

Let N2 = 100↑(↑(…(↑(↑)↑)…)↑)↑100 with N1 layers in my second extended up-arrow notation.

Let N3 = N1→{{…{{N2}}…}} with N1^N2 layers in my extended chained arrows.

Let N4 = f(GN1+N2*N3(37352545625)) in my extended G-function.

Let N5 = AN3(N4) in my extended Ackermann function.

My number is I(I(I(I(I(987654+N1)*N2)^N3)↑↑N4)↑↑↑N5) using your I function.

Now the number is J(I(I(I(I(I(987654+N1)*N2)^N3)↑↑N4)↑↑↑N5))

Oh. I made that post before you finished the J function.

Yeah.

excelguru wrote:

How my second extended up arrow notation works:

After ↑(↑)↑, arrows build up to form more ↑(↑)↑'s, and eventually a long string of ↑(↑)↑'s turns into ↑(↑)(↑)↑. More (↑)'s form between until they collapse to (↑↑), and then (↑)'s and eventually (↑↑)'s form between the ↑'s. Finally the parentheses contain other parentheses.

Now the next step is to define a new type of parenthesis, that turns into (↑(↑(…(↑(↑)↑)…)↑)↑). This is ((↑)). Then after ((↑)) comes ((↑↑)), then ((↑↑↑)), then ((↑(↑)↑)), then ((↑(↑)↑↑)), then ((↑(↑)↑(↑)↑)), then ((↑(↑)(↑)↑)), then ((↑(↑↑)↑)), then ((↑(↑(↑)↑)↑)), then ((↑((↑))↑)), and we finally make it to (((↑))). Then there's ((((↑)))), (((((↑))))), and so on.

Of course we can continue with ([↑](↑)[↑]), ([↑](↑↑)[↑]), ([↑]((↑))[↑]), ([↑]([↑](↑)[↑])[↑]), ([↑][↑](↑)[↑][↑]), ([↑↑](↑)[↑↑]) - but are notations are just getting more and more complicated, so that's where I will stop.

This is getting too complicated

More extensions to my first extended up-arrow notation. I will call it EGEUAN and the second one EG2EUAN.

The next type of bracket after {}_(↑) is not {}_(↑↑), but {}_(↑)(1). Then there's {}_(↑)(2), {}_(↑)(↑), and finally we can define {}_(↑↑). Then continuation goes similarly to EG2EUAN where {}_(↑↑↑) expands to {}_(↑↑)(↑↑)…(↑↑)(↑↑). The limit is {↑}_({↑}_(…({↑}_({↑}))…)).

What comes next?

I am not sure. What I am sure is that the next operator after all that expands to a{↑}_({↑}_(…({↑}_({↑}))…))a with b {↑}'s. For now let's have it be ↑^.

a↑^b
a↑^↑b
a↑^↑↑b

Then all the usual extensions until:

↑^↑^…↑^↑^

And as usual we need a new operator. Let's have it be (↑)↑↑(↑).

a(↑)↑↑(↑)b = a↑^↑^…^↑^↑a with b arrows - I chose this unusual choice because ↑^↑^…^↑^↑ resembles an exponent tower

a(↑)↑↑↑(↑)b = a(↑)↑↑(↑)(↑)↑↑(↑)…(↑)↑↑(↑)(↑)↑↑(↑)a with b (↑)↑↑(↑)'s

Then there are special hyper-operators being applied to (↑)'s so the limit is:

(↑)(↑)…(↑)(↑)↑↑(↑)(↑)…(↑)(↑)

So I will make (↑↑)↑↑(↑↑) be the next operator, then (↑↑↑)↑↑(↑↑↑) will expand to (↑↑)(↑↑)…(↑↑)(↑↑)↑↑(↑↑)(↑↑)…(↑↑)(↑↑), then have nests. Note that this is not likely to be what my actual next extension will be. I don't know what it will be, but it won't be this.

K function

The K function, which implements the * function from here and the G,H,I, and J function, is defined as:

K(1) = **….**(J(I(H(G(G(H(I(J(*(1000010000100001000010000))))))))))) where there are ****(*****(********(*******(100000^1000000)))) *'s
K(N) = TREE(**….**(J(I(H(G(G(H(I(J(*(K(N-1))))))))))))) where there are ****(*****(********(*******(K(N-2)^K(N-1))))) *'s
And the number for this one is K(K(K(K(K(K(K(K(K(K(10000000))))))))))

Will an L function ever come out?

My “under construction” page is finally completed! It is weak arrow notation! Featuring X's.

0^100 = zonarygol = 0
1^100 = unarygol = 1
2^100 = binarygol
3^100 = ternarygol
4^100 = quaternarygol
5^100 = quinarygol
6^100 = senarygol
7^100 = septarygol
8^100 = octalgol
9^100 = nonalgol
10^100 = decimalgol (same as a googol)
11^100 = undecimalgol
12^100 = duodecimalgol
13^100 = tredecimalgol
14^100 = quattuordecimalgol
15^100 = quindecimalgol
16^100 = hexadecimalgol (or sexdecimalgol)
17^100 = septendecimalgol
18^100 = octodecimalgol
19^100 = novemdecimalgol
20^100 = vigesimalgol
21^100 = unvigesimalgol
26^100 = sexvigesimalgol
27^100 = septenvigesimalgol
30^100 = trigesimalgol
40^100 = quadragesimalgol
50^100 = quinquagesimalgol
60^100 = sexagesimalgol
70^100 = septuagesimalgol
80^100 = octogesimalgol
90^100 = nonagesimalgol
99^100 = novemnonagesimalgol
100^100 = centesimalgol
1000^100 = thousesimalgol
1000000^100 = millionsimalgol
1000000000^100 = billionsimalgol
…

(10^100)^100 = googesimalgol
((10^100)^100)^100 = googesimalgolesimalgol
(((10^100)^100)^100)^100 = googesimalgolesimalgolesimalgol
…

Let's have Ax(…) be A(…)(x-1). Then we can have A(…)(…). If it is of the form A(x)(…), treat the second () as if it were the Ackermann function.

A(3)(1,1) = A(3)(0,A(3)(1,0)) = A(3)(0,A(3)(0,1)) = A(3)(0,A(3)(1)) = A(3)(0,A(3,3,3)) = A(3)(triliad) = Atriliad+1(3) = Atriliad(3,3,3)

But A(3,3)(3,3) cannot expand to A(3,3)(2,A(3,3)(3,2)), it has to expand to A(2,A(3,2)(3,3))(3,3).

Now for a new series of numbers:

Duzeriad = A()()
Du-uniad = A(1)(1)
Duduliad = A(2,2)(2,2)
Dutriliad = A(3,3,3)(3,3,3)
Dutetriad = A(4,4,4,4)(4,4,4,4)
…

Stay tuned for when there can be multiple (…)'s!

(the -1 is necessary because the (…) which comes from the subscript has 1 as its default value, and the Ackermann function has 0 as its default value, and the (…) will behave like the Ackermann function, so -1 converts 1 to 0)

Super Graham notation

Have you seen my Graham notation? Well I am going to extend it.

1. a↑[b]c:1 = a↑[b]c
2. a↑[b]c:#:1 = a↑[b]c:#:b
3. a↑[b]c:d = a↑[a↑[b]c:(d-1)]c
4. a↑[b]c:0 = b
5. a↑[b]c:#:0 = 1
6. a↑[b]c:#:d:e = a↑[b]c:#:(a↑[b]c:#:d:(e-1))

Also, did you know G(n) = 3↑↑↑↑3:n? Also, G2(n) = 3↑↑↑↑3:n:n, G3(n) = 3↑↑↑↑3:n:n:n, and in general Gx(n) =
3↑↑↑↑3:n:n:…:n:n with x n's.

excelguru wrote:

10^100 = decimalgol
.

Nope. It's a googol.

ev3coolexit987654 wrote:

excelguru wrote:

10^100 = decimalgol
.

Nope. It's a googol.

Yes I knew that.

Now the number is J↑↑↑J(J(I(I(I(I(I(987654+N1)*N2)^N3)↑↑N4)↑↑↑N5))) where J↑[n]↑J(N) is J↑↑↑[J(n)↑↑↑↑↑↑241242243244245:J(N):J(n)]J(N)
(Note the difference between N and n) Anything in square brackets is means “repeat ↑ this many times”

Graham's numberumpus!

Now let's have G1,2(n) = G_G_…G_Gn(n)(n)…(n)(n) with n n's

G2,2 repeats G1,2
G3,2 repeats G2,2
G4,2 repeats G3,2
G5,2 repeats G4,2

And then G1,3(n) = G_G_…G_G_n,2(n),2(n),2…,2(n),2(n)

Then there's G1,4(n), G1,5, and so on. Now let's have f(n) = G_1,G_1,…G_1,G1,n(n)(n)…(n)(n) with n n's. My number is
f(45676345665434564).

ev3coolexit987654 wrote:

Now the number is J↑↑↑J(J(I(I(I(I(I(987654+N1)*N2)^N3)↑↑N4)↑↑↑N5))) where J↑[n]↑J(N) is J↑↑↑[J(n)↑↑↑↑↑↑241242243244245:J(N):J(n)]J(N)
(Note the difference between N and n) Anything in square brackets is means “repeat ↑ this many times”

What is ↑↑↑[n]? I guess ↑[3n].

excelguru wrote:

ev3coolexit987654 wrote:

Now the number is J↑↑↑J(J(I(I(I(I(I(987654+N1)*N2)^N3)↑↑N4)↑↑↑N5))) where J↑[n]↑J(N) is J↑↑↑[J(n)↑↑↑↑↑↑241242243244245:J(N):J(n)]J(N)
(Note the difference between N and n) Anything in square brackets is means “repeat ↑ this many times”

What is ↑↑↑[n]? I guess ↑[3n].

You could say that.

excelguru wrote:

n = f(45676345665434564).

Improvement: *(1) = Gn, n↑[n]n(n)
* is a function
*(n) = G*(n-1), *(n-1)↑[*(n-1)]*(n-1)(*(n-1))
**(n) = *(*(n))
***(n) = **(**(n))
… and so forth. My number is [insert *(1000) *'s here](****(***(**(*(**(***(10000000))))))).

ev3coolexit987654 wrote:

excelguru wrote:

ev3coolexit987654 wrote:

Now the number is J↑↑↑J(J(I(I(I(I(I(987654+N1)*N2)^N3)↑↑N4)↑↑↑N5))) where J↑[n]↑J(N) is J↑↑↑[J(n)↑↑↑↑↑↑241242243244245:J(N):J(n)]J(N)
(Note the difference between N and n) Anything in square brackets is means “repeat ↑ this many times”

What is ↑↑↑[n]? I guess ↑[3n].

You could say that.

excelguru wrote:

n = f(45676345665434564).

Improvement: *(1) = Gn, n↑[n]n(n)
* is a function
*(n) = G*(n-1), *(n-1)↑[*(n-1)]*(n-1)(*(n-1))
**(n) = *(*(n))
***(n) = **(**(n))
… and so forth. My number is [insert *(1000) *'s here](****(***(**(*(**(***(10000000))))))).

Improvement:

**(n) = *(*(…(*(*(n)))…)) with n *'s
***(n) = **(**(…(**(**(*(n))))…)) with n **'s
****(n) = ***(***(…(***(***(**(*(n)))))…)) with n ***'s
…

My number is **…**(100) where there are *[*(1000)](****(***(**(*(**(***(10000000))))))) *'s (remember what square brackets mean) (this is the new * functions, even though it looks identical to your number).

More improvement:
**(n) = *(*(…(*(*(n)))…)) with n↑↑**(n-1) *'s
***(n) = **(**(…(**(**(*(n))))…)) with n↑↑↑***(n-1) **'s
****(n) = ***(***(…(***(***(**(*(n)))))…)) with n↑↑↑↑****(n-1) ***'s
…

See a pattern? My number is **…**(271828184161803331415)^31415926535897932238462643383279502884197169399375105820974944 where there are *************[***(*(100))](***(*****(*****************(*************(*(1357924680314159265358979))))))

ev3coolexit987654 wrote:

More improvement:
**(n) = *(*(…(*(*(n)))…)) with n↑↑**(n-1) *'s
***(n) = **(**(…(**(**(*(n))))…)) with n↑↑↑***(n-1) **'s
****(n) = ***(***(…(***(***(**(*(n)))))…)) with n↑↑↑↑****(n-1) ***'s
…

See a pattern? My number is **…**(271828184161803331415)^31415926535897932238462643383279502884197169399375105820974944 where there are *************[***(*(100))](***(*****(*****************(*************(*(1357924680314159265358979))))))

What does n↑↑**(n-1) mean?

excelguru wrote:

ev3coolexit987654 wrote:

More improvement:
**(n) = *(*(…(*(*(n)))…)) with n↑↑**(n-1) *'s
***(n) = **(**(…(**(**(*(n))))…)) with n↑↑↑***(n-1) **'s
****(n) = ***(***(…(***(***(**(*(n)))))…)) with n↑↑↑↑****(n-1) ***'s
…

See a pattern? My number is **…**(271828184161803331415)^31415926535897932238462643383279502884197169399375105820974944 where there are *************[***(*(100))](***(*****(*****************(*************(*(1357924680314159265358979))))))

What does n↑↑**(n-1) mean?

Oh. It means n↑↑(**(n-1)). I thought you invented a new function without explaining it.

excelguru wrote:

excelguru wrote:

ev3coolexit987654 wrote:

More improvement:
**(n) = *(*(…(*(*(n)))…)) with n↑↑**(n-1) *'s
***(n) = **(**(…(**(**(*(n))))…)) with n↑↑↑***(n-1) **'s
****(n) = ***(***(…(***(***(**(*(n)))))…)) with n↑↑↑↑****(n-1) ***'s
…

See a pattern? My number is **…**(271828184161803331415)^31415926535897932238462643383279502884197169399375105820974944 where there are *************[***(*(100))](***(*****(*****************(*************(*(1357924680314159265358979))))))

What does n↑↑**(n-1) mean?

Oh. It means n↑↑(**(n-1)). I thought you invented a new function without explaining it.

I assume *[n](1) = 1, right?

By the way, this is my 100th forum post