10/23/2013, 12:26 PM

I was wondering about the following

x^x = e^e^e

x^x^x = e^e^e^e

...

x^^n = e^^(n-1)

if we solve for x we get

(i have little time for research at the moment , but i will give a brute

estimate , which might even be quite exact. )

e^e $$ x

take elog on both sides : e $$ elog(x)

e^e^e $$ x^x

take elog on both sides : e^e $$ elog(x) * x

take elog again : e $$ elog(x) + elog(elog(x))

keep increasing tower hight and taking elogs :

e^e $$ elog(x^x) + elog(elog(x^x))

= e^e $$ elog(x) * x + elog(elog(x) * x)

= e^e $$ elog(x) * x + elog(elog(x)) + elog(x)

= e^e^e $$ elog(x^x) * x^x + elog(elog(x^x) * x^x)

= e^e^e $$ elog(x) * x * x^x + elog(elog(x) * x * x^x)

= e^e^e $$ elog(x) * x * x^x + elog(elog(x)) + elog(x) + elog(x) * x

= e^e^e $$ elog(x)[1 + x + x^(x+1)] + elog(elog(x))

now replace '$$' with '=' and solve for real x > e :

=> q estimated around q = x = 6,6568558380496

in the limit

e^^(n+1) grows slower than 6,6568558380496^^n.

I have confidence in my digits BECAUSE it is known that the convergeance is superexponential.

However we can continue

6,6568... ^^(n+1) = x_2 ^^n

etc

and we end up with a sequence : x_0 = e , x_1 = x = 6.65... , x_2 , x_3 , ...

How fast does this x_n grow ??

it seems x_n is the superfunction of y^^(n-1) = x^^n where n goes to oo but is well approximated by small n FOR A SINGLE STEP AT LEAST.

the behaviour of x_n though troubles me.

this is clearly base change.

But what is the behaviour of the super of y^y^y^y = x^x^x^x^x ?

or even y^y^y^y = x^x^x^x^x ?

is there a known series expansion for y^^(n-1) = x^^n ??

numeric overflow seems like an often encountered problem.

it is easy to show x_n < C x_(n-1)^x_(n-1).

But does x_n then grow faster / slower or equal to exponential ??

I know Lambert W and the Taylor series for a power tower.

Maybe a certain limit with those would help.

x^x = e^e^e

x^x^x = e^e^e^e

...

x^^n = e^^(n-1)

if we solve for x we get

(i have little time for research at the moment , but i will give a brute

estimate , which might even be quite exact. )

e^e $$ x

take elog on both sides : e $$ elog(x)

e^e^e $$ x^x

take elog on both sides : e^e $$ elog(x) * x

take elog again : e $$ elog(x) + elog(elog(x))

keep increasing tower hight and taking elogs :

e^e $$ elog(x^x) + elog(elog(x^x))

= e^e $$ elog(x) * x + elog(elog(x) * x)

= e^e $$ elog(x) * x + elog(elog(x)) + elog(x)

= e^e^e $$ elog(x^x) * x^x + elog(elog(x^x) * x^x)

= e^e^e $$ elog(x) * x * x^x + elog(elog(x) * x * x^x)

= e^e^e $$ elog(x) * x * x^x + elog(elog(x)) + elog(x) + elog(x) * x

= e^e^e $$ elog(x)[1 + x + x^(x+1)] + elog(elog(x))

now replace '$$' with '=' and solve for real x > e :

=> q estimated around q = x = 6,6568558380496

in the limit

e^^(n+1) grows slower than 6,6568558380496^^n.

I have confidence in my digits BECAUSE it is known that the convergeance is superexponential.

However we can continue

6,6568... ^^(n+1) = x_2 ^^n

etc

and we end up with a sequence : x_0 = e , x_1 = x = 6.65... , x_2 , x_3 , ...

How fast does this x_n grow ??

it seems x_n is the superfunction of y^^(n-1) = x^^n where n goes to oo but is well approximated by small n FOR A SINGLE STEP AT LEAST.

the behaviour of x_n though troubles me.

this is clearly base change.

But what is the behaviour of the super of y^y^y^y = x^x^x^x^x ?

or even y^y^y^y = x^x^x^x^x ?

is there a known series expansion for y^^(n-1) = x^^n ??

numeric overflow seems like an often encountered problem.

it is easy to show x_n < C x_(n-1)^x_(n-1).

But does x_n then grow faster / slower or equal to exponential ??

I know Lambert W and the Taylor series for a power tower.

Maybe a certain limit with those would help.