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