My Number is Bigger! (again)

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SirGabriel
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Re: My Number is Bigger! (again)

Postby SirGabriel » Sat Jun 27, 2015 8:38 pm UTC

-sqrt(813.63) * ((tan(ln(sin(log(e)))))^(π^π))
Not sure why Google and Wolfram Alpha are interpreting this differently from the way I intend it to be interpreted, or how to make them interpret it the way I intend it, but the way I'm doing the order of operations I get about 10195.14

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WarDaft
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Re: My Number is Bigger! (again)

Postby WarDaft » Sat Jun 27, 2015 11:57 pm UTC

Where BB(n) is the maximum number of shifts of a 2-symbol n-state Turing Machine:

BB(3)^2 = 11449
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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 1:25 am UTC

5[0]/8 = 12500
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Re: My Number is Bigger! (again)

Postby mike-l » Sun Jun 28, 2015 5:57 am UTC

Vytron wrote:
username5243 wrote:So, each number in the x,0 series seems to be about the cube of the last one, or about exponential growth.


Um, the cube of 6 is 216, while we get 10195 which is more than ^5ing.



Well he's right asymptotically, it's just that the +1 has a large effect for small n. Specifically, if you look at what power of n,0 gives you (n+1),0, the number approaches 3 very very fast (It starts out 5.2, 3.9, 3.5, 3.4, then the numbers are too big to continue, but you can use the following to see that it continues on towards 3)

n, grows like n^3, but there are some extra stuffs, which become less relevant as n gets large, but all the n+1's in place of n's make it seem faster at the start. In particular, n, < (n+2)^3. (n+1 would work except it's not strict for n=0)

n,0 is n+1 applications of the comma operator to n+1.

n+1, < (n+3)^3
(n+1,), < ((n+3)^3)^3 = (n+3)^9
((n+1,),), < ((n+3)^9)^3 = (n+3)^27

inductively we get that n,0 < (n+3)^3^(n+1), which is asymptotically somewhat less than n^n^(n/2) but close enough. (The exact same argument shows that n^3^(n+1) is a lower bound)

Now back to the original claim, to answer what power of x gives y, you do ln (y) / ln (x). To see what power of n,0 gives (n+1),0 we thus do:
ln((n+1),0)/ln(n,0) < ln ((n+4)^3^(n+2)) / ln (n^3^(n+1)) = 3 ln (n+4)/ ln (n), and this rather quickly goes to 3. [the same argument gives 3 as a lower bound by switching where the +2 got applied)
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 10:09 am UTC

Thanks mike. So I gather the function would have a much faster growth if instead of what I'm doing I simply did f(n)=n^4 (i.e. there's some number n by which n,0 < n^4.) The n,0 function does indeed have cubic growth.

mike-l wrote:n, grows like n^3, but there are some extra stuffs, which become less relevant as n gets large


Yes. At n=14612 the difference is already negligible (3 119 965 528 208 368 Vs. 3 119 880 122 529 561 - the former is the result of 2,0 or (146121,) while the later is the result of 146121^3, so the "+2n^2+2n+1" thing can indeed be dropped off safely once n gets big enough; I wonder if I could simplify the function to some (n+x)^3, as it seems it'd be easy to make it easily grow faster and have the same effect on small numbers with some big enough x)

The idea here is I'm never really going to use big numbers as inputs, as by then it's better to just use more 0s at the right of the ,

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((17.1,),)


Spoiler:
(2,0)/((17.1,),)
=
(2,0)/(6205.051,)
=
(2,0)/239064979025.99518
=
3119965528208368/239064979025.99518
=
13050.700863505033

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 10:42 am UTC

I actually did evaluate 2,0 and got about 3.12*10^15. Good to know we're on the same page.

2*4[0] = 20000
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Re: My Number is Bigger! (again)

Postby SirGabriel » Sun Jun 28, 2015 10:55 am UTC

22222

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 10:56 am UTC

5[0]/4 = 25000

Remember, n[0] = 10^n
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 11:08 am UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((15.8,),)


Spoiler:
(2,0)/((15.8,),)
=
(2,0)/(4975.472,)
=
(2,0)/123268440823.59148
=
3119965528208368/123268440823.59148
=
25310.33496783923

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 11:12 am UTC

26*3[0] = 26000
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Re: My Number is Bigger! (again)

Postby SirGabriel » Sun Jun 28, 2015 11:13 am UTC

2^2^2^2

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 11:15 am UTC

I guess that's 2^(2^(2^2)) = 65536, since ((2^2)^2)^2 is 256, much smaller than 26000.

Okay, 2*5[0]/3 ~ 66666.67
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 11:31 am UTC

username5243 wrote:I guess that's 2^(2^(2^2)) = 65536


Yes, ^s are resolved from right to left when no order is specified.

Also, it seems I've been making some mistakes in my calculations, as apparently (2*n)^2 is 4n^2, and I should be using 2n^2 :shock:

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((14.6,),)


Spoiler:
(2,0)/((14.6,),)
=
(2,0)/(3568.656,)
=
(2,0)/45473402656.23551
=
3119965528208368/45473402656.23551
=
68610.77786050709

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 11:35 am UTC

5*5[0]/7 = 71428.57
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 11:41 am UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((14.5,),)


Spoiler:
(2,0)/((14.5,),)
=
(2,0)/(3499.125,)
=
(2,0)/42867346539.17383
=
3119965528208368/42867346539.17383
=
72781.86732078748

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 11:43 am UTC

3*5[0]/4 = 75000
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 11:58 am UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((14.4,),)


Spoiler:
(2,0)/((14.4,),)
=
(2,0)/(3430.504,)
=
(2,0)/40394941720.1487
=
3119965528208368/40394941720.1487
=
77236.5399069793

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 12:07 pm UTC

8*4[0] = 80000
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SirGabriel
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Re: My Number is Bigger! (again)

Postby SirGabriel » Sun Jun 28, 2015 12:16 pm UTC

10^sqrt(24.040292)

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 12:20 pm UTC

Hey, that's about 80000.010885 according to my calculator.

I'm going to have to go bigger to make things harder on Vytron...

5*5[0] = 500000
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 12:30 pm UTC

username5243 wrote:Hey, that's about 80000.010885 according to my calculator.

I'm going to have to go bigger to make things harder on Vytron...

5*5[0] = 500000


Um, I guess what is hard is making this hard for this method :mrgreen:

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((11.5,),)


Spoiler:
(2,0)/((11.5,),)
=
(2,0)/(1809.375,)
=
(2,0)/5930151728.880859
=
3119965528208368/5930151728.880859
=
526119.0051872701

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 12:33 pm UTC

Okay. I wasn't going to jump from eighty thousand to 3.12 quadrillion...

6[0] = 1000000
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 12:42 pm UTC

And I can say this is the least I can overshoot (as in, it would take considerably more effort to overshoot by a lower amount)

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((10.6,),)


Spoiler:
(2,0)/((10.6,),)
=
(2,0)/(1437.936,)
=
(2,0)/2977300860.3740735
=
3119965528208368/2977300860.3740735
=
1047917.4509143728

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 12:44 pm UTC

2^20 = 1048576
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 12:55 pm UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((10.5,),)


Spoiler:
(2,0)/((10.5,),)
=
(2,0)/(14 00.125,)
=
(2,0)/2748658566.908203
=
3119965528208368/2748658566.908203
=
1135086.6076166837

username5243
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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 1:05 pm UTC

Let's speed this up...

7[0] = 10000000
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 1:17 pm UTC

But... but...

username5243 wrote:Also, try not to go too big too fast.


...

Well, I guess having to go from 10.5 to 8 is not too fast.

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((8,),)


Spoiler:
(2,0)/((8,),)
=
(2,0)/(657,)
=
(2,0)/284458006
=
3119965528208368/284458006
=
10968105

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 1:20 pm UTC

Jumps from a million to ten million are okay since we're already pretty used to it. It was mainly for bigger numbers.

8[0]/2 = 50000000
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 1:48 pm UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((6.6,),)


Spoiler:
(2,0)/((6.6,),)
=
(2,0)/(388.816,)
=
(2,0)/59083513.70743448
=
3119965528208368/59083513.70743448
=
52806025.44489128

username5243
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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 1:48 pm UTC

Do you have a table of values for x, or something?

8[0] = 10^8
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 2:10 pm UTC

username5243 wrote:Do you have a table of values for x, or something?


Nah, I'm making each calculation manually by abuse of wolframalpha.com and Big Number Calculator (for top speed.)

For some reason this turns out much easier for me than dividing by 2 even numbers and multiplying by 3 and adding 1 to odd numbers on the Collatz game...

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((6,),)


Spoiler:
(2,0)/((6,),)
=
(2,0)/(301,)
=
(2,0)/27452706
=
3119965528208368/27452706
=
113 648 742

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 2:27 pm UTC

9[0] = 10^9

In about 15 posts we'll pass 2,0!
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 2:35 pm UTC

Unless someone goes insane, of course.

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((4.5,),)


Spoiler:
(2,0)/((4.5,),)
=
(2,0)/(141.625,)
=
(2,0)/2881062.884765625
=
3119965528208368/2881062.884765625
=
1082921703.8982396

username5243
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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 2:47 pm UTC

10[0] = 10^10
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 2:57 pm UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((3.3,),)


Spoiler:
(2,0)/((3,3,),)
=
(2,0)/(65.317,)
=
(2,0)/287326.8571880129
=
3119965528208368/287326.8571880129
=
10858593445.606138

username5243
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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 3:03 pm UTC

11[0] = 10^11
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 3:25 pm UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((2.3,),)


Spoiler:
(2,0)/((3,3,),)
=
(2,0)/(28. 347,)
=
(2,0)/24443.098955923004
=
3119965528208368/24443.098955923004
=
127641979187.43622

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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 3:33 pm UTC

12[0] = 10^12
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Vytron
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Re: My Number is Bigger! (again)

Postby Vytron » Sun Jun 28, 2015 4:11 pm UTC

Reminder: 2,0=3 119 965 528 208 368

Send:
(2,0)/((1.6,),)


Spoiler:
(2,0)/((1.6),)
=
(2,0)/(13.416,)
=
(2,0)/2802.5432872960005
=
3119965528208368/2802.5432872960005
=
1113262208063.3867

username5243
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Re: My Number is Bigger! (again)

Postby username5243 » Sun Jun 28, 2015 4:13 pm UTC

13[0] = 10^13
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