As far as I can tell my original solution is not conclusive. Instead of 8, what about 9, or 10, or N? The N**(1/N) will always find you a value but not all values are a correct solution to the infinite tetration of X. If the value you get falls outside of the Euler limits discussed previously it is not valid.
Than, also only as far as I can tell, my solution using the Lambert W function can also find you solutions that are outside the Euler limits.
So, the final proof depends on understanding how Euler came up with those limits. Which I do not.
The problem with the whole x=n**(1/n) function is that x rises only until n=e and then falls, so there are two n's that give the same x. Infinite tetration of x won't give back any n > e.
Thanks Virand, I didn't notice that calculator before.
This gets curiouser and curiouser...
Electrodude, I and others here concluded that the infinite tetration of X will equal 8 when X = 8^(1/8) = 1.29683955465100
Then we find that this may not be a valid solution because Euler showed that to be valid it must be in the range e^-e to e^(1/e).
That is OK. That range is about 0.06598803 to 1.44466786 and our solution is sitting within that. We have a valid solution.
Then we find we can find this same solution using the Lambert W function. See above. All is well.
Now we have that infinite tetration calculator. Which also warns that valid solutions should be within the range above.
So we plug in our valid solution 1.29683955465100
What to we get out?
1.462501
Oops! What has gone wrong here?
Well, in my post describing the solution via the Lambert W function I said that the online W calculator returned two results: -2.079442 and -0.380148
Taking the first one and completing the calculation gave us 8. The answer we want.
What happens if we take the second value of W instead?
Lo, we get -0.380148 / -Math.log(1.2968395546510096) = 1.462500358410287
The the same solution as the online infinite tetration calculator.
This is all very weird. Maths is broken. Or...I might conclude that there is indeed two solutions, I see nothing to suggest either solution is wrong. Euler says it's OK.
Why do I deal with it?
I have to bridge the time until I finally have the P2 as a chip.
That's why I've collected a few facts.
A solution I have not yet.
:cool:
Whoever invented this tetration to 8 puzzle was very sneaky.
They must have known that if infinite tetration has Euler limits on its input then it also has Euler limits on its output which are its range from 1/e to e.
They must have also known just how much we would be tantalized by so many mirages of 8.
The 8 actually does "mean something" and I suspect it is a point of divergence or anti-solution, because:
Put 7.9 at the top of the tower, it converges to 1.4625
Put 8 at the top of the tower, it won't stay long
Put 8.1 at the top of the tower, it diverges to infinity
1.2968^7.9 <7.9
1.2968^8 =8 (almost)
1.2968^8.1 >8.1
Experiment I can't do much now because I left without laptop:
x=8**(1/8)
v=some value between 1 and 10.
Iterate many times: v=x**v
Seems to always run away and hide from 8.
If we can somehow invert it to Converge on 8 then maybe we can have another infinite tetration function that works for all of the "second solutions" like 8, or else we just got fooled again.
Whoever invented this tetration to 8 puzzle was very sneaky.
That has been bugging me since I heard of it.
I first saw the problem in the video by Mathologer linked to in my previous post: http://forums.parallax.com/discussion/comment/1387545/#Comment_1387545
Mathologer is actually Professor Burkard Polster at Monash University in Melbourne, Australia. For sure he knows all about the Euler limits. In the video he casually mentions this problem in passing and describes it as "one of his favorite equations".
So the question in my mind is why is this a favorite? What's with the number 8 in particular? Is there something special about that 8 or not? Does the 8 actually mean something very special here, as you suggest, or not?
By the way: Surely if you set x=8**(1/8) and v to some value, say 7, then you are calculating a totally different thing than theroblem asks for.
Comments
I'm groping in the dark here but:
As far as I can tell my original solution is not conclusive. Instead of 8, what about 9, or 10, or N? The N**(1/N) will always find you a value but not all values are a correct solution to the infinite tetration of X. If the value you get falls outside of the Euler limits discussed previously it is not valid.
Than, also only as far as I can tell, my solution using the Lambert W function can also find you solutions that are outside the Euler limits.
So, the final proof depends on understanding how Euler came up with those limits. Which I do not.
The site with the Lambert W calculator also has this infinite tetration calculator. Can it say "8"?
http://www.had2know.com/academics/power-tower-iterated-exponentiation-tetration.html
This gets curiouser and curiouser...
Electrodude, I and others here concluded that the infinite tetration of X will equal 8 when X = 8^(1/8) = 1.29683955465100
Then we find that this may not be a valid solution because Euler showed that to be valid it must be in the range e^-e to e^(1/e).
That is OK. That range is about 0.06598803 to 1.44466786 and our solution is sitting within that. We have a valid solution.
Then we find we can find this same solution using the Lambert W function. See above. All is well.
Now we have that infinite tetration calculator. Which also warns that valid solutions should be within the range above.
So we plug in our valid solution 1.29683955465100
What to we get out?
1.462501
Oops! What has gone wrong here?
Well, in my post describing the solution via the Lambert W function I said that the online W calculator returned two results: -2.079442 and -0.380148
Taking the first one and completing the calculation gave us 8. The answer we want.
What happens if we take the second value of W instead?
Lo, we get -0.380148 / -Math.log(1.2968395546510096) = 1.462500358410287
The the same solution as the online infinite tetration calculator.
This is all very weird. Maths is broken. Or...I might conclude that there is indeed two solutions, I see nothing to suggest either solution is wrong. Euler says it's OK.
The graph on the Lambert function calculator page: http://www.had2know.com/academics/lambert-w-function-calculator.html suggests why this is.
When we calculate W(-ln(x)) we are working on the negative side of the graph and can see that there are two solutions.
In conclusion, the infinite tetration of x will can in fact give back 8. Until such time that someone proves why not
I have to bridge the time until I finally have the P2 as a chip.
That's why I've collected a few facts.
A solution I have not yet.
:cool:
The fourth tetration of 2 is 65,536.
Your function returns "inf".
Sorry I stopped reading at that point.
They must have known that if infinite tetration has Euler limits on its input then it also has Euler limits on its output which are its range from 1/e to e.
They must have also known just how much we would be tantalized by so many mirages of 8.
The 8 actually does "mean something" and I suspect it is a point of divergence or anti-solution, because:
Put 7.9 at the top of the tower, it converges to 1.4625
Put 8 at the top of the tower, it won't stay long
Put 8.1 at the top of the tower, it diverges to infinity
1.2968^7.9 <7.9
1.2968^8 =8 (almost)
1.2968^8.1 >8.1
Experiment I can't do much now because I left without laptop:
x=8**(1/8)
v=some value between 1 and 10.
Iterate many times: v=x**v
Seems to always run away and hide from 8.
If we can somehow invert it to Converge on 8 then maybe we can have another infinite tetration function that works for all of the "second solutions" like 8, or else we just got fooled again.
I first saw the problem in the video by Mathologer linked to in my previous post: http://forums.parallax.com/discussion/comment/1387545/#Comment_1387545
Mathologer is actually Professor Burkard Polster at Monash University in Melbourne, Australia. For sure he knows all about the Euler limits. In the video he casually mentions this problem in passing and describes it as "one of his favorite equations".
So the question in my mind is why is this a favorite? What's with the number 8 in particular? Is there something special about that 8 or not? Does the 8 actually mean something very special here, as you suggest, or not?
By the way: Surely if you set x=8**(1/8) and v to some value, say 7, then you are calculating a totally different thing than theroblem asks for.