## Count with PSS expressions of limited length

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PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(0,0)(1,0)(0,0)(0,0)(0,0)(0,0)[10] = {2}{1}14 = 28×228 = 7 516 192 768 ≈ 7.5×109 ≈ E9.88

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(0,0)(1,0)(0,0)(1,0)[10] = {2}{1}{1}10 = 40 * 240 ~ 4.4 * 1013 ~ EE1.135

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

EE1.135

You mean F2.055

At any rate, here are the full decimal form of the last few numbers:
1 048 576
20 971 520
92 274 688
402 653 184
1 744 830 464
7 516 192 768
43 980 465 111 040

(of-course I won't be able to keep that going for much longer )

(0,0)(1,0)(1,0)(0,0)(1,0)(0,0)(1,0)(0,0)[10] = {2}{1}{1}{0}10 = 44×244 = 774 056 185 954 304 ≈ 7.74×1014 ≈ E14.89 ≈ F2.069

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(0,0)(1,0)(0,0)(1,0)(0,0)(0,0)[10] = {2}{1}{1}12 = 48 * 248 ~ 1.35×1016 ~ F2.082

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

That was 13 510 798 882 111 488

And now:

(0,0)(1,0)(1,0)(0,0)(1,0)(0,0)(1,0)(0,0)(1,0)[10] = {2}{1}{1}{1}10 = {2}80 = 80×280 =
= 96 714 065 569 170 333 976 494 080 ≈ 9.67×1025 ≈ E25.985 ≈ F2.151

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)[10] = {2}{2}10 = 10240 * 210240 ~ 3.61 * 103086 ~ EEE0.543 = F2.543

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

What? No decimal expansion? It's just:

360957072599898384090605039595548830355368154562216663658407757882686159634992633498094902235890317951559353911247855349486172666633657483467656588012709208748182598356636384334399545815230549708652121305780944094607687051610452806545332430422287685702892997010122003214335508082605878763758507043556411021131882440599131490156862577439953257503054733504014811497383642697438777679619716430794701581449591679689093164315539356126569695803705455174307071116910866061066273442321577547102316976919997191759869752309980772581503326976891743409084980526800964685721621214255852907801943482545725312697422823198175086555263549935207147167916759294630345216830044858194310497469585643388793214920683723361413054467883378587623186830388026709606288329594972154022376673485434803813697555172939568214923110437549731129443639611126859852738744989346253589445842275335089709057738999107362704933446827567946264837762487245074658161789298542273863070037238565810037387697951019945803617801257085790876437756594716845933151615975229353907304463152808968984861025962180160659334221678013914000051722776924934944720336623246536410349671563495582840474975535382609467326519059383049608782570197824570156088538044006148380691681254757288694690489481905743711844181842852717743112266019072319762196328421011040303954345027726782941861882418169484142610931473711982979105056933005658672562620510427959810381446455477346563284940509928834430832475909325288857376410879046828625223279640651753897335374339878482047069769788820613993720652178330715228866933918010269286022686582658559288492858064139725178753096222327775043137725961893906665695825694925440377077920743638601960837791117641595877476961307332573071625842682355292133107640045483246956029669323559670280750415727846152425844774850423021498101974337867585724795113066625866510161810993557764641190807624788591668574563114300977843921380910589544188835912484131356109086668225057069880775806656833318513865539755205915006035377306676434213935959195464072399104805532814456133113489977561952607282641277077600317528478934753582181943660217291661472783057094767504962805464905449619694327203383583442885922628270574456545002062883179784228297808225974006038595728828620619879518842170350704751184638636449217039082062288188714696175654411105225993151788599812851415654264842837422304284010164509705982693049731471323109170157173297417942125904467996480337653121462763247430028011322300406163756468357136334325433159782564464868518326705844968870557751649823818717425250349192461845919023290072836614736734772217186495720772586702700364156747687912174603961965388970385558595245890211448129278043639109041815070649122062005392558094379119925775750476208325991144162271899491238942998087295383317029962614619645583924749228199962075775515843940323113975912791834585885958615365879524094710625916041626044389326756510449876143147527025058780988709896628161732334426568002262550585352954698777482484398843216975680153506461202889572423805010320468864660836910084980649358090891381521055951158218869435922794230497899829410896991347525797191088330506240

But let's stop with that before it gets out of hand...

(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)[10] = {2}{2}{0}10 = {2}(11×211) = {2}22528 = 22528×222528 = 1022528×log(2)+log(22528) ≈ 106786 = E6786 ≈ F2.583

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

I leave the crazy stuff to you.

(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(0,0)[10] = {2}{2}12 = 49152 * 249152 ~ 8.3 * 1014800 ~ EEE0,62 = F2.62

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

I leave the crazy stuff to you.

Good call

(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(0,0)(0,0)[10] = {2}{2}{0}{0}{0} = {2}(13×213) = {2}106496 = 106496×2106496 = 10106496×log(2)+log(106496) ≈ 1032063.52 ≈ 3.3×1032063 ≈ E32064 ≈ EEE0.654 = F2.654

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(1,0)[10] = {2}{2}20 = 20971520 * 220971520 ~ 7.876 * 106313063 ~ EEE0.83 = F2.83

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(1,0)(0,0)[10] = {2}{2}22 ≈ 1027777457 ≈ F2.87

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)[10] = {2}{2}{2}10 ~ {2}(3.61 * 103086) ~ EE3086 ~ F3.543

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

We can use Robert's Munafo's power tower notation to write the approximations now:

N pt x = a power tower of N 10's with an x on top.

(0,0)(1,0)(1,0)(1,0)[10]= {3}10 = {2}1010 ≈ 9pt3086
(that's 10^10^10^10^10^10^10^10^10^3086)

This notation should remain useful for the next dozen entries or so.

(the above can also be approximated as F10.54 or as G2.0043, the latter being the standard [letter]x form with 2<x<10)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)[10] = {3}11 = {2}11[11] ~ 10pt6786 ~ F11.58 ~ G2.01

I need to re-familiarize myself with the letter notation, that took me way too long to double-check.

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(0,0)[10] = {3}12 = {2}1212 ≈ 11pt14801
(≈ F12.62 ≈ FEE0.0418 ≈ FF1.0418 = FFE0.0178 = G2.0178)
Last edited by PsiCubed2 on Wed Jun 05, 2019 10:54 am UTC, edited 1 time in total.

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(0,0)(0,0)[10] = {3}13 = {2}13[13] ~ 12pt32064 ~ F13.65 ~ G2.023

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(0,0)(0,0)(0,0)[10] = {3}14 = {2}1414 ≈ 13pt69054
(≈ F14.68 ≈ FEE0.0670 ≈ FF1.0670 = FFE0.0282 = G2.0282)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(0,0)(0,0)(0,0)(0,0)[10] = {3}15 = {2}15[15] ~ 14pt147968 ~ F15.71 ~ G2.0325

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)[10] = {3}20 = {2}2020 ≈ 19pt6313063
(≈ F20.83 ≈ FEE0.1202 ≈ FF1.1202 = FFE0.0493 = G2.0493)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(0,0)[10] = {3}22 = {2}22[22] ~ 21pt27777457 ~ F22.87 ~ G2.054

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(0,0)(0,0)[10] = {3}24 = {2}2424 ≈ 23pt121 210 694)
(≈ F24.91 ≈ FEE0.1450 = FF1.1450 = FFE0.059 = G2.059)
(≈ F25 = 10↑↑25)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(0,0)(0,0)(0,0)[10] = {3}26 = {2}26[26] ~ 25pt525246316 ~ F26.94 ~ G2.063

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(0,0)(1,0)[10] = {3}40 = {2}4040 ≈ 39pt13239439221689 ≈ 40pt13
(≈ F41.05 ≈ FEE0.208 = FF1.208 ≈ FFE0.082 = G2.082)
(≈ F41 = 10↑↑41)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

If I didn't screw it up

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(0,0)(1,0)(0,0)[10] = {3}44 = {2}44[44] ~ 43pt(2.33*1014) ~ F45.06 ~ G2.086

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

Looks good to me.

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)[10] = {3}{2}10 = {3}{2}10 ≈ {3}10240 ≈ 10240pt3086
(≈ F10241.54 ≈ FE4.01 ≈ FEE0.603 = FF1.603 ≈ G2.205)
(≈ F10242 = 10↑↑10242)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

Help me out please. How is this step calculated?

{3}10240 ≈ 10240pt3086

Are you just substituting 10 as the base instead of 2 in the chain?

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

It's exactly the same calculation we've done for the previous dozen numbers or so:

{3}44 = {2}4444 ≈ 43pt(2.33×1014) ≈ 43pt(1014) = 44pt14

{3}10240 = {2}1024010240 ≈ 10239pt103186 = 10240pt3186

The trickiest part in both calculations, is the second step. But since you clearly know how to do it for n=44, I'm not sure why you're having a hard time with n=10240.

At any rate, here is a general approximation for {2}kn when k>2,n>3:

{2}kn ≈ (k-1)pt(n×2n×log(2)+log(n)+n×log(2)+log(log(2)))

For k=3, the righthand formula is simply the result of calculating {2}{2}{2}n the hard way and using the rules of logarithms to convert all the 2's into 10's.

For k>3, we use the fact that for a large enough n (say n>F4), 10n is a very good approximation of {2}n=n×2n. Thanks to this fact, the expression to the right of the "pt" (in the approximation) doesn't depend on k.

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

PsiCubed2 wrote:But since you clearly know how to do it for n=44, I'm not sure why you're having a hard time with n=10240.
Because I'm lazy and use wolfram alpha and one of the steps was too much for it.

PsiCubed2 wrote:For k>3, we use the fact that for a large enough n (say n>F4), 10n is a very good approximation of {2}n=n×2n.
That's what I guessed, thanks for confirming.

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)[10] = {3}22528 = {2}22528[22528] ~ 22528pt6785.435 ~ F22529.58 ~ G2.2145

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

Well, I'm even lazier.

I no longer do any calculations at all. I just take a peek at the previous values and adjust them as needed.

For example, {3}10240 and {3}10 are both equal to {2}....{2}10 (with 10241 and 10 {2}'s respectively). We've already calculated that {3}10 is about 9pt3086. So we can write immediately, without any further calculation, that {3}10240 = 10240pt3086.

Using this trick, I'll do the next one:
(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(0,0)[10] = {3}{2}12

Looking up the thread, I see that:
(a) {2}12 = 49152
(b) {3}12 ≈ 11pt14801 ≈ F12.62

So I can write immediately:

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(0,0)[10] = {3}49152 = 49152pt14801 ≈ F49153.62
(≈G2.223, which is the only point I've used a calculator)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

Okay, let me try that one:

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(1,0)[10] = {3}{2}20 ~ 20971520pt6313063 ~ F20971521.83 ~ G2.27

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

Yup.

Now for a pretty big step:
(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)(1,0)[10] = {3}{3}10 ≈ {3}(9pt3086) ≈ 10↑↑(9pt3086) = F(9pt3086) = FF10.54 ≈ FF10 = G3.00
( ≈ 10↑↑↑3)

And I propose another way to write the numbers form now on:

[ a,b ; x ] = FaEbx with 0<=a,b<=8 integers and 10<=x<1010

Any number between 10 and G10 can be written uniquely as such an expression. Note that:

[ 0,b ; n ] = b pt n
[ 1,0 ; n ] = n pt 1 = Fn

So the previous number could be written as [ 1,0 ; 20971521.83 ]

FF10.54 = [ 2,0 ; 10.54 ]

(funny how we've jumped entirely over all the numbers of the form [ 1,b ; n ] for 1<=b<=8)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

Next step's gonna be even bigger, no? No room left at {3}, you have to move to {4} level.

(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)(1,0)(0,0)[10] = {3}{3}11 ~ F({3}11) ~ FF11.58 = [ 2,0 ; 11.58 ] (~ G3.0115)

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

Exactly.

[ a,b,c ; x ] = GaFbEcx

(0,0)(1,0)(1,0)(1,0)(1,0)[10] = {4}10 = {3}1010 = {3}9{3}10 ≈ F9({3)}10 ≈ F9F10.54 = F1010.54 ≈ F111.01 ≈ G11.00 = 10↑↑↑11 = [ 1,0,0 ; 11 ]
( ≈ H2.003)

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(1,0)(0,0)[10] = {4}11 = {3}1111 ~ F10({3}11) ~ F1111.58 ~ G12.0115 ~ [ 1,0,0 ; 12 ]

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(1,0)(0,0)(0,0)[10] = {4}12 ≈ G13 = [ 1,0,0 ; 13 ]

Kinda boring to write the numbers like this... How about allowing the entries of the approximation array to be higher than 8? Then we'll get:

{4}12 ≈ F11({3}12) ≈ F11(11pt14801) = [ 11,11 ; 14801 ]

And the previous few numbers in the same format:
{3}{2}{1}10 ≈ 20971520pt6313063 = [ 20971520 ; 6313063 ]
{3}{3}10 ≈ [ 1,9 ; 3086 ]
{3}{3}{0}10 ≈ [ 1,10 ; 6785 ]
{4}10 ≈ [ 9,9 ; 3086 ]
{4}11 ≈ [ 10,10 ; 6785 ]

And it looks like in general:
{4}n ≈ [ n-1,n-1 ; (the value after the 'pt' we've shamelessly stolen from the previous {3}n calculation) ]

Sabrar
Posts: 42
Joined: Tue Nov 03, 2015 6:29 pm UTC

### Re: Count with PSS expressions of limited length

But doesn't that defeat the purpose of having a concise and unique representation?

(0,0)(1,0)(1,0)(1,0)(1,0)(0,0)(0,0)(0,0)[10] = {4}13 ~ [ 12,12 ; 32064 ]

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

(0,0)(1,0)(1,0)(1,0)(1,0)(0,0)(0,0)(0,0)(0,0)[10] = {4}14 ~ [ 13,13 ; 69054 ] ~ [ 1,0,0 ; 15 ]

Hmmm... You're right.

To make this variation unique, we need to define an exact cut-off point at which to stop expanding the expression. How about this:

1. We the write the number as <some letter>x for 2<x<10.
2. We expand the number with the usual rules of letter notation.
3. We stop the expansion when either of the following two things happen:
(a) The expansion was completed (i.e. no letters remained in the expression)
(b) The last letter is "E" and the number after it is bigger than 10.
4. We convert the resulting expression into the [ a,b,c,...,n ; x ] format, and that's our new standard form.

Posts: 49
Joined: Thu Jun 25, 2015 10:43 am UTC

### Re: Count with PSS expressions of limited length

I semi-arbitrarily decide to check in here and see that this exists...

(0,0)(1,0)(1,0)(1,0)(1,0)(0,0)(1,0)[10] = {4}{1}10 = {4}20 ~ G21 ~ [ 1,0,0 ; 21 ]
This is a signature, in case you didn't notice.

Nayuta_Ito
Posts: 5
Joined: Fri Jun 07, 2019 1:43 am UTC

### Re: Count with PSS expressions of limited length

It seems still OK to post here (in order not to post on a dead ten-year-old thread)

(0,0)(1,0)(1,0)(1,0)(1,0)(0,0)(1,0)(0,0)[10] = {4}{1}11 = {4}22 ~ G23 ~ [ 1,0,0 ; 23 ]

I wonder how much will this "sweet time" continues.

PsiCubed2
Posts: 12
Joined: Tue Apr 02, 2019 8:55 pm UTC

### Re: Count with PSS expressions of limited length

How come so many of you guys came here in such a short notice?

Is there a new googology forum I'm not aware of? If so, please do tell