Many many years ago I conjectured the following :

Consider exp iterations of a starting value \( y=0+si \) for \( 0<s<2 \).

It seems for all such s , after some iterations we get close to 0.

So we search for positive real x > 1 , such that

\( exp^{[x]}(y)=0 \)

or it gets very close.

I believe an upper bound for x is

\( bound(x)=1+exp(s^2+s+1/s) \)

I gave it the funny name conjecture 666 because by the formula above :

\( bound(2)=666.141.. \)

Notice the formula does not extend correctly to values such as \( s=\pi \) since that sequence will never come close to zero.

A sharper bound is probably attainable.

But how ?

Is there an easy way ?

For 1/2<s<2 this might be achievable by computer search ? And/or calculus ?

But is there an easy or short proof ?

Notice that taking derivatives of exp(exp(... is not easier than computing exp(exp(... so it seems hard to shortcut the problem.

I was inspired to share this idea of mine here because of memories of some people on sci.math.

What do you guys think ?

Regards

tommy1729

Consider exp iterations of a starting value \( y=0+si \) for \( 0<s<2 \).

It seems for all such s , after some iterations we get close to 0.

So we search for positive real x > 1 , such that

\( exp^{[x]}(y)=0 \)

or it gets very close.

I believe an upper bound for x is

\( bound(x)=1+exp(s^2+s+1/s) \)

I gave it the funny name conjecture 666 because by the formula above :

\( bound(2)=666.141.. \)

Notice the formula does not extend correctly to values such as \( s=\pi \) since that sequence will never come close to zero.

A sharper bound is probably attainable.

But how ?

Is there an easy way ?

For 1/2<s<2 this might be achievable by computer search ? And/or calculus ?

But is there an easy or short proof ?

Notice that taking derivatives of exp(exp(... is not easier than computing exp(exp(... so it seems hard to shortcut the problem.

I was inspired to share this idea of mine here because of memories of some people on sci.math.

What do you guys think ?

Regards

tommy1729