An extremely long power tower

In summary, the conversation discusses the concept of a "power tower" with an extremely large number, which is too massive to be calculated in decimal notation. The conversation also mentions the Ackermann function and the impossibility of expressing such a large number in decimal digits in a reasonable amount of time. The conversation concludes with the acknowledgement that the number cannot be expressed and a mention of ultrafinitism.
  • #1
golden3159
2
0
An extremely long "power tower"

Hey, this isn't really a homework or classwork question. My professor always puts questions that he finds interesting on our notes. This really got my attention, I've been trying to see if it's possible to write a computer program to solve it.

However, I've come to the conclusion that this number is simply too large to be calculable in decimal notation. I was just wondering if you could tell me if I've done the right things.

First of all here is the information for the problem:
http://img62.imageshack.us/img62/4931/problemy.th.png [Broken]

And the problem itself
[PLAIN]http://img687.imageshack.us/img687/1779/49195211.png [Broken]

(Sorry for using so many pictures instead of typing this out, my browser is having some issues with using the toolbar items above). Anyways, after working this out, the conclusion that I come to is that I need to keep a power of 2's, in the following fashion:
[PLAIN]http://img687.imageshack.us/img687/5981/part2s.png [Broken]

I have to have a height of more than 65,000 on the tower! As far as I am aware, this would produce a number FAR too massive to compute. Am I on the right track and is there any possible way to even obtain a decimal notation of the above problem? Thanks.

EDIT: Just a side thought, is there any relation to this and the Ackermann function at all?`
 
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  • #2


The number you describe is [itex]wow(4)[/itex], i.e.[itex]f_4(4)[/itex] in the Ackermann hierarchy - the Ackermann function itself being [itex]f_\omega[/itex].

There is no chance of an exact computation in decimal digits in meaningful time, nor would there be any way of physically holding or displaying the digits.
 
  • #3


golden3159 said:
is there any possible way to even obtain a decimal notation of the above problem?

I understand you probably realize this is too massive to be expressed in decimal notation (and even too massive to be expressed reasonably as it is in the power tower form!) but just watch at how fast the tower power grows in decimal notation so your hopes of expressing this can be destroyed into fragments of shattered dreams... :tongue:

[tex]2^2=4[/tex]

[tex]2^{2^2}=2^4=16[/tex]

[tex]2^{2^{2^2}}=2^{16}=65,536[/tex]

[tex]2^{65,536}\approx 10^{19,728}[/tex]

Only 4 applications and the resulting number is 19728 digits long!
 
  • #4


Presumably "A" stands for Ackerman.
 
  • #5


http://en.wikipedia.org/wiki/Ultrafinitism" [Broken]
 
Last edited by a moderator:
  • #6


Looking back at my original post, I am not sure why I even asked if it was possible to express in the first place. I should have known better. I guess I just wanted to make sure I had arrived at the proper notation at least. Thanks for the answers guys.
 

1. What is an extremely long power tower?

An extremely long power tower is a structure used to harness energy from the sun by reflecting and concentrating sunlight onto a central receiver, which then converts the energy into electricity.

2. How does an extremely long power tower work?

The power tower works by using a series of mirrors, called heliostats, to reflect and concentrate sunlight onto a central receiver. The receiver then heats a fluid, typically molten salt, which is used to generate steam and power a turbine to generate electricity.

3. What are the advantages of an extremely long power tower?

Some advantages of an extremely long power tower include its ability to generate large amounts of electricity, its efficiency in converting solar energy into electricity, and its ability to store energy for use when the sun is not shining.

4. What are the challenges of using an extremely long power tower?

One of the main challenges of using an extremely long power tower is its high cost of construction and maintenance. Another challenge is the need for a large amount of land to build the power tower and its associated infrastructure.

5. Are there any environmental concerns with using an extremely long power tower?

While an extremely long power tower is a renewable energy source, there are some environmental concerns to consider. These include the potential for land and habitat disruption during construction, as well as the use of water for cooling the power tower and potential impacts on local wildlife.

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