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I was just wondering if it is possible to have pi{pi up arrows}pi? in other words is this a valid expression or are knuth up arrows only valid for positive integers?
A further question I have is on whether or not real numbers can be used as up arrow values for example can we have phi^pi up arrows?
Going beyond that is it possible to have complex numbered up arrows?
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The trimerticus is equal to {10,12,1010,12} in my array notation.
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===Up to .
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With the numbers on May 26th, we can learn that PEGG will suapass googol in between June 4th and July 3rd. Well... I mean June 4th 2017 and July 3rd 2019. Why is the range so large? Because the growing speed of PEGG varies by 100 times according to DJIA.
Now Here's the poll:
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In this 3rd part of the analysis of my array notation I am going to analyze the {a,b,cd,k} part of my notation.
Before I explain what's happening with {a,b,cd,k} I need to go over the rule for {a,b,cd} again as it is essential for the next rule.
{a,b,cd}= {a,b,c} recursed d times for d>0
If d=0 {a,b,cd} = {a,b,c} as no recursions are being applied to {a,b,c}
Otherwise {a,b,cd}= {a,b,c} recursed into the b slot d times.
This rule allows numbers such as the puny Grahams number to be expressed as {3,6,364} is exactly equal to grahams number.
Now onto {a,b,cd,k} where {a,b,cd} is recursed k times.
Here is the rule for this section of the notation:
{a,b,cd,k} = {a,b,cd} recursed k times for k>0
If k=0 {a,b,cd,k} = {a,b,cd}. Otherwise {a,b,…
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This is my attempt to define UNAN for noninteger values of the base and iterator. (I'm not going to try with numbers inside the brackets.)
I was partly inspired by PsiCubed's letter notation, which some of the definitions I used are based on.
note: a[#]b is always increasing only for a > 2. For a e^{1/e}. For smaller values of a, the tetration is generally known to either converge or alternate between 0 and 1. I haven't looked into what happens for pentation and higher, but I'm fairly sure for a > 1 it will still be an increasing function, just converging to one number.
Twoentry arrays with the first entry in the brackets > 0 expand similar to the last rule above. Now, we just need to handle cases of type a[0,n]b.
 a[0,n+1]b = a[1,n]a^{b} for 0 < b …
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While playing around with my letter notation, I stumbled across this interesting number:
Je (in my letter notation) ≈ 10↑↑10↑↑10↑↑10↑↑x
where x is a power tower of 43 740 904 567 012 530 171 588 786 071 630 323 521 909 315 122 877 770 449 840 384 426 342 tens topped by a 4322.
Or for an even more precise approximation:
x is a power tower of 43 740 904 567 012 530 171 588 786 071 630 323 521 909 315 122 877 770 449 840 384 426 341 tens topped by the following 4322digit number:
59384455081745432082663190267515052102323029594214457015975670574841674734511616140010660811092628207996935879293056374599839144438235560095525648074936698443220094817407713693991543667020897613431766576718909477129039744791383057596483365851268713799365808400577405032008…
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Expression written in notation is exactly equal to output of function of fast growing hierarchy indexed by ordinal number generated by Buchholz's function.
What inspired me:
Chronolegends's Egg Notation
Deedlit's notation
Buchholz's function
Fastgrowing hierarchy
ab corresponds to \(f_b(a)\) where \(f_b\) is a function of fastgrowing hierarchy
To the right of the sign "" :
1) () corresponds to 1, (()) corresponds to \(\omega\) and (...) always corresponds to a countable ordinal number,
2) \(()_b\) corresponds to \(\Omega_b\) where \(\Omega_b=\aleph_b=\psi_b(0)\) denotes bth uncountable ordinal,
3) \((...)_b\) corresponds to \(\psi_b(...)\) where \(\psi_b\) denotes Buchholz's function.
notation allows to obtain ultimatively short ruleset for w…
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Poopillion equal is H(infinity*two)
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Penisillion or kinfinillion or tinfinitillionillion or infinitillionillionillion or equal is H(H(Infinity))
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This post is the 2nd out of the series of posts detailing how my notation works this time we're doing {a,b,cd}
To format a rule here is an explanation of what is happening:
{a,b,cd}= {a,b,c} recursed d times in other words the answer you get from recursing it will go into the b slot.
For example {2,2,22}= 2*2=4 for one recursion so the 1st recursion will produce an answer of 2^^2=4 the next recursion produces 2^^2=4.
The rule for this part of the notation is:
{a,b,cd}= {a,b,c} recursed d times for d>0
if d=1 {a,b,cd}= {a,b,c}
Otherwise {a,b,cd}= {a,b,c} recursed into the b slot d times.
To illustrate this last point here is an example:
{3,4,33}= 3^^3= 0 recursions
3^(3^27)3= 1 recursion
3^(3^(3^27))3= 2 recursions.
3^(3(^(3^(3^27))))3 = 3 recur…
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This blog will be the first of a series of blog posts outlining the workings of my notation in more detail starting with this one that will explore the first level of my notation mainly the {a,b,c} part outlined in my blog titled "My next attempt at an array notation."
We begin by writing the definition of what {a,b,c} is: {a,b,c} = a_(b2) up arrows c.
To outline this concept I'll start b=1 then move up the b's so as to fully explain what {a,b,c} does.
{a,1,c}= a+c simple right nothing highly drastic here just simple addition this forms the basis of the rest of the notation as everything else can be derecursed to this level.
{a,2,c}= a*c.
{a,3,c}= a^c
{a,4,c}=a^^c
{a,5,c}=a^^^c
{a,6,c}=a^^^^c
{a,7,c}=a^^^^^c
{a,8,c}=a^^^^^^c
As you can see {a,b,c}= …
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I have a slight complaint to make because when I tried to put my Ternary number on this wiki from my own website it got deleted in a matter of hours can anyone explain why this is?
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This version of The Alpha Function has been rewritten to use Javascript in Google Sheets. The code is available for anybody to use or copy as they like. This code replaces my last version which used VBA in Microsoft Excel.
The function code is still based on The S Function (Version 2), with a growth rate of \(f_{svo}(n)\).
Version 9 has been completely rewritten to use Javascript in Google Sheets. A link to the first draft Google Sheet file is available here:
First Draft Google Sheet File
Version 9 has also been 'recalibrated' to allow an input parameter range from 0 to 100,000 that should be more interesting. The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated by Javascript Code to…
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Hi. I'm thinking of adding several numbers. But was hoping for some guidance because this is my first foray into posting here ("Long time "listener" first time "caller"! ... aka noob! :)
Here are the numbers I'd like to add:
2^(2^63), i.e., 2^9223372036854775808; which is obviously uncomputable, although WAlpha tells us there are 2776511644261678567 digits! :)
and, by contrast:
(2^2)^63, i.e., 4^63, which is a mere: 85070591730234615865843651857942052864
I'd like to contrast these if possible. I guess that I could link the pages to one another.
The other numbers I'm interested in posting are a bit harder to classify. They are the numbers you get by taking the decimal point out of common irrationals like Pi and e, so they are uncomputable infin…
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The ternary number is equal to [3,3,33,3,3#3,3,3(3,3,3)]= [3,3,33,3,3#3,3,3#3,3,3#3,3,3.......] (3,3,3) #3,3,3's.
The notation for numbers like this is explained in my previous post.
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We start this with {#,#,1,2} having level phi(1,0,0,0)
({#,#,1,2})#= phi(phi(1,0,0,0),0,1)
({#,#,1,2})##= phi(phi(1,0,0,0),1,0)
({#,#,1,2}){{#,#,1,2}+1}#= phi(phi(1,0,0,0)+1,0,0)
{{#,#,1,2},#,1,2}= phi(1,0,0,1)
{#,#+1,1,2}= phi(1,0,0,w)
&(1)= phi(1,0,0,w^2)
&(2)= phi(1,0,0,w^2*2)
&(#)= phi(1,0,0,w^3)
#*^#= phi(1,0,1,0)
#*^##= phi(1,0,1,1)
#*^#^#= phi(1,0,1,w)
#*^#*^#= phi(1,0,1,phi(1,0,1,0))
#*^^#= phi(1,0,2,0)
#*^^##= phi(1,0,3,0)
#*^^^#= phi(1,1,0,0)
#*^^^^#= phi(1,2,0,0)
#*{#}#= phi(1,w,0,0)
#*{#,#,1,2}#= phi(2,0,0,0)
Beyond this point, it becomes ambigous what comes next, for example how #*{#,#+1,1,2} evualates. Everything beyond this point are just guesses.
#*{#,#+1,1,2}#=phi(2,0,0,w)
#*&(1)#=phi(2,0,0,w^2)#**^#=phi(w,0,0,0)
#/^#=phi(1.0,…
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Well, it looks like the textbook is missing a chunk. It should be done in a few weeks.
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Last time I got to [a,b,cd,k#p#q] in my notation where there are two elements in the chain after the k (p and q) this time instead of having n+1 terms in the array after the k you can have multiple terms. For example [a,b,cd,k#(3)]= [a,b,cd,k#p#q#r]the r in this array works the same way the q would work in the expression [a,b,cd,k#p#q] the only difference is instead of having [a,b,cd,k#p] arrays of [a,b,c] you have [a,b,cd,k#p#q] arrays of [a,b,c] you can extend this idea to fit any number of arrays denoted as [a,b,cd,k#(n)]
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I can only update this chart on Wednesdays, Fridays, Satuadays, and Sundays. If you want to see the value for today, please check PsiCubed2's Current Value Of PEGG.
This is the first "age" of PEGG. In this age, X is added by 0.01*(The hundredth place of the previous day's DJIA closing price) and PEGG is 10^X.
Therefore, PEGG grows approximately exponentially.
date what number DJIA on
previous day ended with hundredth place of the
DJIA closing value (n) 0.01*n^2 X PEGG value
1
4
19
19
21
21
22
22
33
58
72
79
114
165
239
295
301
331
588
2570
11220
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And currently, nothing interesting will happen at least until May 20th. Now 28th. 
The PEGGLog is now maintained by Nayuto Ito:
http://googology.wikia.com/wiki/User_blog:Nayuta_Ito/PEGG_detailed_log
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The crux of Letter Notation is the following array notation:
(a,b,c,...,k)x
To write the rules for expanding such arrays, we'll use the following shorthand:
Y  represents any string of numbers seperated by commas (can also be empty).
Z  represents a string of zeros seperated by commas (can also be empty).
n,k  nonnegative integers.
x  a nonnegative number (not necessarily an integer)
i  the integer part of x
f  the fractional part of x
And now for the rules. There are 8 of them:
1. For x
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Last time I got to [a,b,cd,k#p] now I'm not going to explain this as I explained it last time. This time I've extended it to greater levels [a,b,cd,k#p#q] works in sort of the same way as [a,b,cs,k#p] except that instead of having [3,3,33,3#3] you would have [3,3,33,3#3] arrays of [3,3,33,3#3] after the k.
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A day or two ago, PsiCubed2 created an evergrowing googolism, or PEGG for short. I think this is a great idea, and I wanted to try out the idea a bit further, and wanted to know what will happen to PEGG in the future. So I decided to simulate it.
Since I don't think I can predict future values of Dow Jones, I slightly modified the definition of PEGG for our simulation purposes. Specifically, rule (a) in the definition is replaced with the following:
(a) We take the (days passed since 20170508)th integer in the result from the Random Integer Generator by random.org, where the minimum value is 1, the maximum value is 10, and the pregenerated randomization from 20170508 is used (you'll need to switch to Advanced Mode to set this). Call it…
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It starts with a C, and ends with a D.
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WELL THIS IS GOOGOLOGY IS DEFINED BY
FFJHDGGNCJVHRFRSFHGJCMKFHJFJTXKUFTJLXHKFJGDCKNGFYJLFIJTDJITFHITDMLHDKTHFKHRDMLYDNTKF FUCK EVERYONE I AM THE BOSS OF THE WORLD AND FUCK EVERYONE WITH THEIR NUMBERS SINCE MINE IS BIGGER AND I AM 4 AND YOU ARE POOPHOLES. I AM 4 JUST LIKE YOU AND I AM RUDE TO YOUR NUMBERS AND RUINING MY LIFE FOR THIS SHIT AND FUCK LATEX FOR SHITTING ME TO BE HACKED TO HAVE A BLACK HOLE SASQUATCH LIGHT YEARS IN DIAMETER COME JUDT TO TORTURE ME. LIST OF GOOGOLOGICAL FUNCTIONS USED BELOW
CHRFKJYDJGRFKLYXKHFYDHKGGGXKGHSYJLFJGRDKL7DJGRFJLXFDHDJ,YZVHMZJVMFBX,UGHRD RWFESMHTDHMRDKLYFKHFDKFKHRDGXGSFXGFZJTXHRXKFRJAGKHDMHF,SNGFARGEGRQRG MEANINGLESS NEVER ENDING EGO GAME OF ONE UPSMANSHIP THAT MAKES PEOPLE RUDE VYFTCJTSHFJSUGHFSKGSNGFNFA…
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Inspired by the infamous Lynz , I hereby define a googolism that changes with time which I'll call Psi's EverGrowing Googolism (PEGG):
1. On May 8, 2017, we set:
X = 0.00
Y = "E"
Z = 0.01
K=1000
PEGG = X = E0.00 = 1
2. Every day after that, we follow this algorithm:
(a) We take the hundredths digit of the Dow Jones Industrial Average at the end of the previous day and call it n. If n=0 then we set n=10. For weekends and other noncommerence days, we use the n from the previous day.
(the reason for using the Dow Jones is to have a public source of pseudorandom numbers).
(c) X+(n^{2}Z) → X
(d) int(X) → PEGG
(for example, if Y is the letter E and X=3.1 then we'll set the new value of PEGG to int(E3.1) = 1258.
(e) If X>10 than we also do the following:
(e1) …
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I am hoping Googology I will be finished by May 10. I have given example quizzes and brief summaries of how to teach the class. The textbook is still incomplete. As of today, I am a third through Unit 2 (Functions and Notations). If I don't finish it by the 10th, it should be done by Monday, the 15th of May.
Thanks, Simon Weston 00:47, May 7, 2017 (UTC)
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Here are some patterns and sequences that I find interesting (some of these I invented my myself).
 The Supremus sequence
This sequence follows this rule : 3X+1^3X+2
{ 1024 ,5764801 , 100000000000 , 3.937376385699×10^15...}
Complete Binary strings
 sequences of the format 111111111111111....
This sequence is founder by using (2^n)1 1=1 11=3 111=7 1111=15 11111=31 111111=63 1111111=127
extended factorial
n!^n1!^n2!^....nn+1! {1,2,36,4.870849395847×10^49...}
tower sequence n^n^n^n...^nn+2^nn+1.
{1,4,10^2.115682024465×10^38...}
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Yottaillion or Yottunitillionillion or Yottmillionillion equal is 10^{3*103*1024+3}3?
3⋅103⋅septillion+3
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[a,b]=a+b.
[a,b,c]=a^(b) c.
[a,b,c (d)] = a,b,c recursed d times.
[a,b,c (d,e)]= [a,b,c (d)] recursed [a,b,c(d)] times for 1 cycle of recursion then repeated again until you have done e cycles of recursion.
[1,2]=3
[2,3,4]=2^^^4=2^^(2^^(2^^2))=2^^(2^^(4))=2^^(2^2^2^2)=2^^(65536).
[3,3,3 (2)] = 3 followed by 3^^^3 up arrows then 3.
[3,3,3(2,5)] = [3,3,3(2)] up arrows for 1 cycle if we call this A then the next 1 will have A number of up arrows between the 3's. In total you have to do [3,3,3(2)] times for 1 complete cycle. So the number of complete cycles is 5 as e=5.
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What is it?
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I have defined rectangles to be recursive squares where squares have a growth rate of f_omega (n) and rectangles have a growth rate of f_omega+1 (n). This regiment is known as the polygon regiment as it is two dimensional in nature it has a growth rate of f_omega2 (n). The next regiment is the 3 dimensional regiment or the block regiment it has a growth rate of f_omega3(n) , this regiment is defined as fw2+n (n) where n is the number of sides in an n sided polygon. For example a cube which is the first member of this regiment has 6 faces so its fw2+6 (n) for a cube. You can probably see that this will be bounded at f_omega3(n). The next regiment follows the same sort of rules the only difference beings its f_omega3+n (n) instead of f_omeg…
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This is an idea I had for a new notation that is not based on arrays (entirely).
I call it Euclidean notation.
square [a,b,c] = a^(b)c or a followed by b up arrows then c.
rectangle = square in a square or recursive squares: This is where you recurse squares to produce even bigger numbers.
This is the first part of the notation later on I'll extend it to incorporate more shapes.
Here is an example of a square and a rectangle: square[ 4,4,4]=4^^^^4. rectangle:4[4,4,4] means that you recurse square[4,4,4] 4 times. Grahams number= rectangle:64[3,4,3] so its still quite small compared to whats next.
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Yesterday I was browsing the main wiki when I came across the article on copy notation invented by wiki user Sponge Tech X. The notation is used to define repeated digits in a number for example 5[2]=55 , this is an example of Hyper mathematics as it is based on concatenation the mathematical operation for gluing two numbers together.
Here is an example of copy notation as well as some extensions on it that I came up with or was inspired by.
1[1]=1
2[2]=22
3[3]=333
4[4]=4444
5[5]=55555
6[6]=666666
7[7]= 7777777
8[8]=88888888
9[9] = 999999999
10[10]=10101010101010101010
Here are some of my extensions to this
2[2[2]] = a chain of 22 2's or 2222222222222222222222.
2[2[2[2]]]= a chain of 2222222222222222222222 2's.
2[2[2[2[2]]]] = a chain of 2[2[2[2]]] 2's.
A…
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The next stage in my notation is this [a,b,cd,k#p] where the #p represents the number of entries into the array after the k. For example [6,6,66,6#6] would mean that their are 6 entries of 6 in the array after the k.
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The world of big numbers can be confusing, so I thought about creating a nice little table that shows the progression of larger and larger numberrealms in some of the simplest notations. I'll compare BEAF, my letter notation, and Denis Maksudov's extended arrows. I'm not using the Fast Growing Hierarchy here even though it is standard, because I fear it will only confuse people. Suffice to say that for the bigger numbers, the fast growing hierarchy is comparable to Maksudov's arrows.
So here we go:
Letter Notation My Array Notation BEAF Arrows Notes
E2 12 {10,2,1} 10↑2 One Hundred = 100
E3 13 {10,3,1} 10↑3 One Thousand = 1000
E4 14 {10,4,1} 10↑4 Ten Thousand = 10,000
E5 15 {10,5,1} 10↑5 One Hundred Thosuand = 100,000
E6 16 {10,6,1} 10↑6 One…
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Last time I got to [a,b,cd] in my notation where [a,b,c] is recursed d times , this time I am going to introduce something that will get it way past what I did with my last notation of which this is an improved version of. [a,b,cd,k] means that you recurse [a,b,cd,k] for some value k [a,b,cd,k1] times. For example [3,5,33,5] means means that you recurse [3,5,33] , [3,5,33]1 times then repeat that 4 times as k is equal to 5.
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This is part 2 of my notation hope you enjoy:
Last time I showed you the first part of the notation mainly [A,^(n2),B] where n is a hyperoperator this time I'm going to show you how to get beyond this.
Suppose that we want to express a number like the Graham Gardner number in this format how can we do it in a way that looks clean?
Yes in fact we can because we can have [3,6,364] this 3^^^^3 recursed 64 times giving us the definition of grahams number.
[A,^(n2),BP] is equal to [A,^(n2),B] recursed P times.
As we have got that definition out of the way we can begin to create some numbers in this format.
 [5,5,55]=5^^^5 recursed 5 times.
 [10,100,1040]=10^(98)10 recursed 40 times.
 [3,6,31000000]=A Forcal in Aarex's Graham generator.
 [500,502,50…

After looking over my notation for a while I started to see that it wasn't going anywhere so I decided to rethink my ideas and start anew.
{A+B}={A,1,B}
The 1 stands for addition.
{A*B}={A,2,B}
The 2 stands for multiplication.
{A^B}={A,3,B}
The 3 stands for exponentiation.
To give some examples of this {3,3,3}=3^3=27 and {2,2,5}=2*5=10.
{A^^B}={A,4,B}
: : {A,400,B}
Or A and B seperated by 398 up arrows.
The rule of this part of the notation is very simple it is mearly just:
A and B seperated by n2 up arrows when n is a Hyperoperator.
{A,^(n2),B}
Now that we have this rule down we can start to think about expressing some numbers in this fashion.
{4,5,6}=4^^^6
{10,10,10}=10^(8)10.
{1000,1000,1000}=1000^(998)1000.
As you can see this rule of having A and B w…
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I wanted to try and make an amazingly large googolism with a unique notation to make my first nonsalad number. I define PEPTO BISMOL BELOW.
(PEPTO(PEPTO(PEPTO(PEPTO(PEPTO(PEPTO(PEPTO(PEPTO(PEPTO(10^100))))))))))
PEPTOS, where PEPTO represents the amount of bits of info needed to exactly calculate to the smallest unit every data set. It is much higher than you think because of the need to calculate every tiny fluctuation and how it changes every other tiny fluctuation. Also needing to factor in everything else in the cosmic event horizon to get it to the Planck units.
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Let me know if theres some specific numbers or another notation you guys would like me to add., i don't promise to add every suggestion but i'll take ideas.
The right side counts the red eggs upon applying the notation to the left side
Improved Egg Notation Analysis Table
iEN FGH
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Last time we looked at the part of my notation that goes like this (a,b,c#[x]{y}{k}[n]) the k is the numerical index of the notation and the n is what you are applying it to (a,b,c#[x]{y}{k}[n]) has an approximate growth rate of fw+k+4(n)fw2(n). For example (a,b,c#[x]{y}{1}[5]) is approximately equal to fw2(5)=fw+5(5) and (a,b,c#[x]{y}{96}[100]) is approximately equal to fw2(100)=fw+100(100). The next stage after this would be to get the notation up to the fw3 to fw^2 level of the fast growing hierarchy. (a,b,c#[x]{y}{k}[n]p) the (a,b,c#[x]{y}{k}[n]) is very familiar to us but the p isn't. I'll give an example then explain what is happening (a,b,c#[x]{y}{1}[5]5) is approximately equal to fw2+5(5)=fw3(5). (a,b,c#[x]{y}…
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Tripencostikollion is equal to (3,f,3#500) in my notation or (3,f,3) recursed 500 times.
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Here is the first number that I am going to define using the notation I have been working ,although it isn't big at least it demonstrates the start of my Notation later on I will give numbers using the later parts of my notation.
(10,Z,10)=10^(50)10 or 10 followed by 50 up arrows then 10.
Z is the limit for what we can do without any recursion.
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