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Crazy conjecture connecting the sqrt(e) and tetrations!
#1
I don't have a powerful enough computer to determine this, but I would like someone to tell me that I'm wrong.

I will let the data speak for itself:

e^(1/e) = 1.444...
Let d = 1/e

Set infinity to be some arbitrarily high number, e.g. 9.99e10000000

Take the following sequences:

...^ (1.4444 + d) ^ (1.4444 + d) ^ (1.4444 + d) ^ (1.4444 + d) ^ (1.4444 + d)

...^ (1.4444 + d^2) ^ (1.4444 + d^2) ^ (1.4444 + d^2) ^ (1.4444 + d^2) ^ (1.4444 + d^2)

...^ (1.4444 + d^3) ^ (1.4444 + d^3) ^ (1.4444 + d^3) ^ (1.4444 + d^3) ^ (1.4444 + d^3)

Continue to increase n toward infinity...
...^ (1.4444 + d^n) ^ (1.4444 + d^n) ^ (1.4444 + d^n) ^ (1.4444 + d^n) ^ (1.4444 + d^n)

Each of these sequences reaches "infinity" after the following iterations:
8, 12, 16, 25, 41, 66, 108, 178, 293, 482, 794 when using (d,d^2,d^3,d^4,d^5,d^6,d^7,d^8,d^9,d^10) respectively.

This looks like a geometric series based close to sqrt(e) = (1.645...).

Perhaps the number of iterations to get to "infinity" approaches sqrt(e)???

Bo, I'm awaiting your superior mathematical intellect.

Ryan
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#2
(04/21/2010, 07:19 PM)rsgerard Wrote: e^(1/e) = 1.444...
Let d = 1/e

Set infinity to be some arbitrarily high number, e.g. 9.99e10000000

I can further generalize this conjecture:

if d= 1/c, for any constant > 1

the infinite tetration of e^(1/e) + d, will reach "infinity" after 1/sqrt© iterations. I can post the data if anyone is interested:

For example, when d=1/10 we reach "infinity" after:
12, 34, 104, 325, 1024 iterations for d=(1/10,1/100,1/10^3,1/10^4)

This series grows at sqrt(10) for each iteration approximately.

Ryan
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#3
(04/21/2010, 07:48 PM)rsgerard Wrote:
(04/21/2010, 07:19 PM)rsgerard Wrote: e^(1/e) = 1.444...
Let d = 1/e

Set infinity to be some arbitrarily high number, e.g. 9.99e10000000

I can further generalize this conjecture:

if d= 1/c, for any constant > 1

the infinite tetration of e^(1/e) + d, will reach "infinity" after 1/sqrt© iterations. I can post the data if anyone is interested:

For example, when d=1/10 we reach "infinity" after:
12, 34, 104, 325, 1024 iterations for d=(1/10,1/100,1/10^3,1/10^4)

This series grows at sqrt(10) for each iteration approximately.

Ryan

Hm, so what you are saying is that

Or at least

where and is the inverse function of

Sounds really interesting, however I have no idea how to tackle.
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#4
(04/22/2010, 12:41 PM)bo198214 Wrote:
(04/21/2010, 07:48 PM)rsgerard Wrote:
(04/21/2010, 07:19 PM)rsgerard Wrote: e^(1/e) = 1.444...
Let d = 1/e

Set infinity to be some arbitrarily high number, e.g. 9.99e10000000

I can further generalize this conjecture:

if d= 1/c, for any constant > 1

the infinite tetration of e^(1/e) + d, will reach "infinity" after 1/sqrt© iterations. I can post the data if anyone is interested:

For example, when d=1/10 we reach "infinity" after:
12, 34, 104, 325, 1024 iterations for d=(1/10,1/100,1/10^3,1/10^4)

This series grows at sqrt(10) for each iteration approximately.

Ryan

Hm, so what you are saying is that

Or at least

where and is the inverse function of

Sounds really interesting, however I have no idea how to tackle.

i noticed that too , very long ago.

perhaps the count till 'oo' is the confusing part.

what if we replace d with -d and count until we reach 'e' (instead of 'oo')

then would the limit also give sqrt© ?

if so , i think we are close to a proof.

or at least arrive at showing these limits depend on earlier conjectured limits ( such as the limit by gottfried )

regards

tommy1729
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#5
i couldnt help noticing that sqrt(e) occurs here , just as it does in my " use sinh " thread.

could there be a link ??

it seems sqrt(e) is the number 3 constant after e^1/e.

1) e^1/e
2) fixpoint e^x
3) sqrt(e)
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#6
(04/22/2010, 12:41 PM)bo198214 Wrote: (...)
Sounds really interesting, however I have no idea how to tackle.

Hmm, I do not really see a good possibility to tackle this. Just want to note that one can rewrite this



where and and then

with some k
Gottfried Helms, Kassel
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#7
(02/28/2011, 02:43 PM)Gottfried Wrote:
(04/22/2010, 12:41 PM)bo198214 Wrote: (...)
Sounds really interesting, however I have no idea how to tackle.

Hmm, I do not really see a good possibility to tackle this. Just want to note that one can rewrite this



where and and then

with some k

Hmm. This is quite an old post.

I remember thinking I know how Gottfried arrived at this.

But I seem to have forgotten now.

Maybe I should have posted my ideas back then.

Could you plz explain Gottfried ?

regards

tommy1729
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#8
(04/22/2010, 12:41 PM)bo198214 Wrote: Hm, so what you are saying is that

Maybe I am wrong but I seem to disagree with that.

Does this not contradict the base change ?

Since the base change dictates that

regards

tommy1729
Reply


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